Whenever the turn rolls back around to me running something hopefully the Ordo Xenos book will be out, and I want my group and I to do a bit of a world building exercise and craft the Inquisitor they will be serving under.

What kind of questions do you think are important/unimportant, and what questions would you ask/want to be asked?

I think one of the most important questions to settle is where they fall on the radical/puritan spectrum. Hashing that out as a group will make sure everyone is on the same page regarding how the group handles forbidden knowledge, objects, etc., as well as what actions are permissible and what will get you executed (the correct answer of course being that anything and everything is permitted as long as you're doing it for the Emperor).

If you want to settle this question in more concrete terms, one of the DH1 books, RH or IH, lists out all of the various Inquisitorial factions, their philosophies, conspiracies, and how they operate. Having the players collectively decide on one will give you a good sense for the kind of game they want to play.

Puritan or Radical?

This would obviously affect the type of missions the players might be sent on, but also what actions the Inquisitor would approve of and which they would censure. A campaign with a Radical Inquisitor might even drift into internal strife as other Inquisitors begin to investigate them and possibly forbidden pet projects.

On the extreme end of the spectrum, the players may end up caught in a black ops war against other Imperial organisations more often than they'd fight the actual enemies of humanity; to some, this can be an interesting prospect, but it also allows for a lot of "grey area" and difficult choices when the Inquisitor is able to present a good reason for why they're doing what they do ("This can help us win the war!").

Militant or Covert?

Apart from influencing the Inquisitor's visual appearance, this too would have an effect both on the type of mission as well as how the players are supposed to start out / approach it. In the broadest sense, it's the difference between a public witch hunt and cloak & daggers. Most Inquisitors fall somewhere in-between, and some covert Inquisitors even use the actions of their militant colleagues as a smokescreen to conduct their own operations in the shadows whilst everyone is distracted by the guy/gal in shining armour laying waste to half the city.

Furthermore, this could provide a hint to the Inquisitor's own background (meaning, what they did before they were recruited) and specialist knowledge gained from such a past. Very likely, it will also affect the type of contacts the Inquisitor has available, and which can in turn be made available (in limited fashion) to the players, from Death Cult assassins specialised on inconspicuous murder to a Chapter of Space Marines standing by to break down the governor's palace doors if need be.

.....I swear I've got something from yonks ago back here......

Aha!

Right. Many years ago, there was a 40k 'narrative wargame' called Inquisitor (AKA The One With The 54mm Miniatures), which was sort of a forerunner to the 'proper RPG' of Dark Heresy. A lot of the Inquisition background today came out of that game, and since your warband leader was (usually) an inquisitor, there was a bit of discussion in one of the issues of Fanatic/Exterminatus/Dark Magenta (the various official and semi-official magazines) about fleshing out an inquisitor.

A suggested list of questions for your character was put together, and some of them are paraphrased below.

1. Is the Inquisitor a Radical, Moderate or Puritan - as Lyata stated. Note that this is tied to questions 2 and 3 - things which an Ordo Xenos inquisitor considers borderline heresy may be far more acceptable to the ordo hereticus, and vice-versa - for example using xenotech against psykers, or psykers against xenos.
2. What Faction, if any, does the Inqusitor belong to - there are a hell of a lot of sub-factions, as CPS stated, which further outline an Inquisitor's overriding goals and/or methods
3. What Ordo, if any, does the Inquisitor belong to - acolytes of an Ordo Xenos inquisitor may still be asked to deal with any sort of foe, but they'll have initially been recruited and equipped with either xenos or xenos influences (like the faceless trade) in mind. It also gives a guideline as to what sort of 'Big Gun' reinforcement characters should be permitted; not everyone gets the speed-dial code for Broadsword Station.
4. Is the Inquisitor subtle or overt - as Lynata stated.
5. Where is the Inquisitor from? An Inquisitor from Askellon, who's grown up around a dying sector, may have a vastly different approach to one who has come from outside the sector and sees its fall as inevitable and the worlds as a resource to be wrung dry before it's too late. Equally, educated and informed as an Inquisitor usually is, there's no substitute for actually being on home ground. If a mission requires you to dig up some of your inquisitor's past, the option is there. Also, this will shape his actual skills - relevant if he takes to the field in person.
6. Does they have any family/friends/allies? Inquisitors tend to operate in loose 'cells' or informal networks based on previous associations, prior inquisitor/interrogator relationships, common interests or foes, etc. A senior acolyte can expect to see and know some of his/her master's inner circle.
7. Does the inquisitor have any enemies/rivals? You may not know who the cult of nine thousand names is. They may not know who you are, but if they find out you work for that Inquisitor , the one who ruined the ceremony of the conjunction...... equally, limited 'wars' between inquisitors and factions break out all the time. Finding out partway through a break-in operation that the facility in question is an inquisitorial archive can cause consternation amongst the acolytes, to say the least.
8. How closely does the inqusitor work with the acolytes? Are they prime agents, like Harlon Nayl/Carl Thonius, who work with their master all the time? Are they one team amongst dozens, with the Inqusititor as the chief/'M'/superintendant at headquarters? Is he nothing more than a voice on the vox, or have they never even met him in person at all? There is something to be said for both sides - the inquisitor gives the GM a pair of hands in play, but it's supposed to be the Acolytes story, not his. Equally, there is something very satisfying (as with the short story in the rulebook) when something serious is going down and the players realise " oh, bugger, you actually came in person for this one?!?!?!?"
9. How much does the Inquisitor keep them informed/supported? Part of the above, really, but still relevant. Some inqusitors will not hand out information which is dangerous to know, obviously, but some take 'need to know' to obsessive levels. Equally, whilst the mechanic for using the inquisitor's influence exists, how readily will he allow them to use it? Are the acolytes provided with the gear they need for a specialised job, and given access to Inquisitor armouries, etc, or do they need to make do with what they've got and beg, borrow or steal anything else.

.....I swear I've got something from yonks ago back here......

Aha!

Right. Many years ago, there was a 40k 'narrative wargame' called Inquisitor (AKA The One With The 54mm Miniatures), which was sort of a forerunner to the 'proper RPG' of Dark Heresy. A lot of the Inquisition background today came out of that game, and since your warband leader was (usually) an inquisitor, there was a bit of discussion in one of the issues of Fanatic/Exterminatus/Dark Magenta (the various official and semi-official magazines) about fleshing out an inquisitor.

A suggested list of questions for your character was put together, and some of them are paraphrased below.

1. Is the Inquisitor a Radical, Moderate or Puritan - as Lyata stated. Note that this is tied to questions 2 and 3 - things which an Ordo Xenos inquisitor considers borderline heresy may be far more acceptable to the ordo hereticus, and vice-versa - for example using xenotech against psykers, or psykers against xenos.
2. What Faction, if any, does the Inqusitor belong to - there are a hell of a lot of sub-factions, as CPS stated, which further outline an Inquisitor's overriding goals and/or methods
3. What Ordo, if any, does the Inquisitor belong to - acolytes of an Ordo Xenos inquisitor may still be asked to deal with any sort of foe, but they'll have initially been recruited and equipped with either xenos or xenos influences (like the faceless trade) in mind. It also gives a guideline as to what sort of 'Big Gun' reinforcement characters should be permitted; not everyone gets the speed-dial code for Broadsword Station.
4. Is the Inquisitor subtle or overt - as Lynata stated.
5. Where is the Inquisitor from? An Inquisitor from Askellon, who's grown up around a dying sector, may have a vastly different approach to one who has come from outside the sector and sees its fall as inevitable and the worlds as a resource to be wrung dry before it's too late. Equally, educated and informed as an Inquisitor usually is, there's no substitute for actually being on home ground. If a mission requires you to dig up some of your inquisitor's past, the option is there. Also, this will shape his actual skills - relevant if he takes to the field in person.
6. Does they have any family/friends/allies? Inquisitors tend to operate in loose 'cells' or informal networks based on previous associations, prior inquisitor/interrogator relationships, common interests or foes, etc. A senior acolyte can expect to see and know some of his/her master's inner circle.
7. Does the inquisitor have any enemies/rivals? You may not know who the cult of nine thousand names is. They may not know who you are, but if they find out you work for that Inquisitor , the one who ruined the ceremony of the conjunction...... equally, limited 'wars' between inquisitors and factions break out all the time. Finding out partway through a break-in operation that the facility in question is an inquisitorial archive can cause consternation amongst the acolytes, to say the least.
8. How closely does the inqusitor work with the acolytes? Are they prime agents, like Harlon Nayl/Carl Thonius, who work with their master all the time? Are they one team amongst dozens, with the Inqusititor as the chief/'M'/superintendant at headquarters? Is he nothing more than a voice on the vox, or have they never even met him in person at all? There is something to be said for both sides - the inquisitor gives the GM a pair of hands in play, but it's supposed to be the Acolytes story, not his. Equally, there is something very satisfying (as with the short story in the rulebook) when something serious is going down and the players realise " oh, bugger, you actually came in person for this one?!?!?!?"
9. How much does the Inquisitor keep them informed/supported? Part of the above, really, but still relevant. Some inqusitors will not hand out information which is dangerous to know, obviously, but some take 'need to know' to obsessive levels. Equally, whilst the mechanic for using the inquisitor's influence exists, how readily will he allow them to use it? Are the acolytes provided with the gear they need for a specialised job, and given access to Inquisitor armouries, etc, or do they need to make do with what they've got and beg, borrow or steal anything else.

These are all pretty good recommendations. I have a few comments to add.

1. I would say this is the most important question to answer as it will have a huge impact on the game. A radical campaign will be very different than a puritan campaign. The players don't even need to line up perfectly with the inquisitor. Could lead to some interesting conflict if the party's views start to differ from their inquisitor. Also worth noting that the PCs may not even know for sure where their inquisitor falls. A puritan may will obviously not gain much by appearingbut a radical may be very careful and hide a lot from anyone but his closest allies making him appear moderate or even puritanical to his lower level agents. Either way the GM should have a clear idea in his head when planning the campaign.

2. Not really important on its own but it could help a lot in answering some of the other questions and for the GM to plan the campaign. This is something starting characters are unlikely to know much about without forbidden lore inquisition.

3. I'd actually say this is one of the less important questions. Most investigations start without knowing much about the nature of the threat and by the time they know its not like they can throw their hands up and say "this isn't my ordo's specialty" and walk out. Mostly useful for knowing what sort of allies or enemies the inquisitor might have. Just like factions, ordos may be something PCs don't even realize exist without a forbidden lore.

4. Very important. Knowing this will determine what sort of orders/missions the GM should have the inquisitor give and could in theory influence the groups play style.

5. Probably don't need more than a vague idea on this. What sector, what type of homeworld and maybe pick a background as well.

6. On top of family/friends/allies, which would likely be relationships that operate more on favors rather than direct control, what resources does he directly control/possess himself. Very important as it has a direct rules effect, what is his influence. Also things like what's the size of his acolyte network, does he have anything like hidden safe houses, does he have any standing militant forces of his own or does he always borrow them, any warp capable ships etc.

7. Obviously pretty important since this can lead directly to who or what the antagonist of the campaign will be. Can also create fun little complications if he has a few powerful rival adeptus or inquisition personnel.

8. Also figuring out a rough personality, at least in relation to how he treats his acolytes. Does he use them like disposable pawns or treat them as valuable allies and brothers in arms.

9. Ties in to my comment in #8. His opinion of his acolyte may influence how much he shares with them.

would add:

10. Vaguely what is the Inquisitor good at and what does he or she look like? Only important if he will be interacting with the party directly. How does he fight? Master swordsmen, or gunner, or not much of a fighter? Social character, knowledge character etc.? Is the Inquisitor a psyker or not? These can also inform what the inquisitor may look like and ties in with #4 as in does he wear plain robes or power armor covered in I's and purity seals and is he obviously armed or does he carry hidden weapons or does he carry no weapons and rely on psychic powers.