As with the previous post (Stark Weenie), these are decks I've come up with after just playing the Core Set for my girlfriend and I so that we know what to buy. Hrakkar Pelt is a good cantrip, and combos well with Street Waif. Pyromancer's Apprentice is a beautiful way to counter pesky locations, Maegi Crone should be all the draw I need once I get going, and the rest should be self-explanatory.

Cards I'm looking at/for would be: The Dragonpit, and any cards that would allow me to cycle my dead/discard pile back in to my deck. Once again, I'd appreciate any suggestions. Thank you for your time.

Characters:

3x Street Waif (King's Landing)
2x Drogon (Core Set)
3x Rhaegal (Core Set)
1x Viserion (Core Set)
3x Herald of the East (A Clash of Arms)
3x King's Landing Assassin (King's Landing)
3x Maegi Crone (Defenders of the North)
1x Ser Barristan Selmy (King's Landing)
3x Pyromancer's Apprentice (King's Landing)

Events:

2x Forever Burning (Core Set)
3x Aegon's Legacy (King's Landing)
2x The Hatchling's Feast (Defenders of the North)
3x Westeros Bleeds (Core Set)
3x True Power (Defenders of the North)

Attachments:

3x Flame-Kissed (Core Set)
3x Bones of a Child (Core Set)
3x Dragon Skull (King's Landing)
3x Hrakkar Pelt (Defenders of the North)

Locations:

3x Eastern Fiefdom (Core Set)
1x Khal Drogo's Tent (Core Set)
3x Summer Sea (Core Set)
1x Lady Daenary's Chambers (Core Set)
3x Flaming-Pitch Tower (A Clash of Arms)
1x Aegon's Hill (King's Landing)
1x Rhaenys's Hill (King's Landing)

Plots:

1x Building Season (Core Set)
1x Valar Morghulis (Core Set)
1x Counting Favors (Core Set)
1x Rule by Decree (Core Set)
1x City of Lies (King's Landing)
1x Fury of the Dragon (A Clash of Arms)
1x Regroup (Kings of the Storm)

Visyrian Spirit if you know she is playing Winter.

If I'm going to do that, would it be safe to run the Kings of Winter Agenda? - and if so, do I need to run the ravens to maintain it? This card looks awesome, by the way.

My bad, its Valyrian Sprirt. No, don't run Winter as Targ, there isnt much of anything to support it and it would waste valuable space in your deck. I use Spirit only if I suspect or know Winter will be played by an opponent, especially a GJ opponent due to all the discard effects. I don't see any reason to ever play Winter as Targ... maybe in the right burn deck you could run Kings of Winter and try to keep a small hand with lots of influence for attachment recursion, figuring the gold penalty for Winter won't be as hard on you as it is on an opponent, but I think it'd be tricky.
I experimented with the idea of a Winter burn deck with a Knights of the Hollow Hill Agenda, but the winter would really only be in there to hurt an opponent a little bit without actually helping me much. If anyone has any thoughts about this or has had any success I would love to hear what works.

This "tricky" sounds intriguing. I'd love to try and build it. Besides the spirit, however, is there anything else I really need to work on with this deck?

Keeping mind that I am not the most experienced player, my thoughts are as follows:
Your current listing is really really heavy on attachments and events and very character-lite. Though this can work you'll probably have trouble drawing characters and you'll end up with a lot of things you cant play; Summer Seas and Pelts and Bones don't do much good if you cant get a character to play them with. Core Khal Drogo is really useful because of his ability, a free character for challenges as long as you manage to win a challenge; 3 Rhaegals is probably 1 too many. Even Illyrio's man would be a good character choice since he pays for himself and can become fodder the same turn, preventing you from having to kill the characters you really care about. Check out the neutral characters on Dabbler for more character ideas.
Personally I don't like having 3 Flame-Kissed since they don't work on any character who already has an attachment or a "no attachments" character. Others will disagree with me and always go with 3 Flame-Kissed, but I suggest just 2. If you want to keep a lot of attachments, definately throw in a Pentos or two. Tears of Lys is a great recursive draw mechanic... win an intrigue challenge, discard it to draw a card, get it back with Lady Daenery's Chamber, repeat. You should probably add a few more locations too, Xaro's is a good draw mechanic if you play with a lot of attachments. I would recommend taking out some of the Westeros Bleeds due to the high influence cost; if you really like Westeros Bleeds, get a Red Keep or two, 3 influence for 3 gold and its hard for an opponent to get rid of since it goes back to shadows instead of being removed from play.

Also, your plots produce very little gold on average - 4-2-3-4-3-5-2 and you have no gold producing locations aside from Drogo's Tent. Most of the time you can't even use your Viserion since you won't have 4 gold, and even if you do, you won't be able to marshall anything. Consider adding Threat From the North to plots (helps burn, but also low gold production), take out Counting Favors or Building Season since both help your opponent. Fear of Winter could also be useful if you need to stall for a round, you can still bring Dragon Skulls and Assassins out of shadows without it counting as playing a card. Although it seems like a good card, City of Lies is low gold production and you really don't have that many Shadows cards so I would get rid of it and think about replacing it with something that makes a lot more gold. Song of Summer is usually an ok choice, 5 gold and 5 initiative even if you're not running Summer (maybe she will be). You have better ways to get cards back than the 2 gold Regroup, though the 2 claim is nice (again, Fear of Winter would be a better choice I think). Rule by Decree is ok but not if you plan on drawing a lot of cards, you are only hurting yourself, Fury is good, Valar is... well, its Valar; nice for clearing the board out but only 2 gold, no claim and another player can usually recover from it just as quickly as you can if not faster unless you've been hammering away at his/her hand.

And remember, you need to be able to claim power! A lot of new players build a deck that looks really good and has good card combos, and only later realize they don't have a good reliable way to win power challenges. Hope this helped!

Agenda:

Knights of the Hollow Hill

Characters: 23

3x Carrion Bird (A Time of Ravens)
3x Samwell Tarly (A Time of Ravens)
3x Pyromancer's Apprentice (King's Landing)
1x Ser Barristan Selmy (King's Landing)
2x Rhaegal (Core Set)
2x Drogon (Core Set)
3x Street Waif (King's Landing)
3x King's Landing Assassin (King's Landing)
1x Xaro Xoan Daxos (Core Set)
2x Valyrian Spirit (A Time of Ravens)

Attachments: 13

2x Tears of Lys (King's Landing)
3x White Raven (A Time of Ravens)
3x Bones of a Child (Core Set)
3x Dragon Skull (King's Landing)
2x Flame-Kissed (Core Set)

Events: 9

1x Forever Burning (Core Set)
2x The Hatchling's Feast (Defenders of the North)
3x Westeros Bleeds (Core Set)
3x Aegon's Legacy (King's Landing)

Locations: 15

1x Lady Daenerys's Chambers (Core Set)
3x Flaming-Pitch Tower (A Clash of Arms)
1x Aegon's Hill (King's Landing)
1x The Dragonpit (King's Landing)
1x The Red Keep (King's Landing)
1x Rhaenys's Hill (King's Landing)
2x Summer Sea (Core Set)
2x Shivering Sea (Kings of the Sea)
3x Eastern Fiefdoms (Core Set)

Plots:

1x Valar Morghulis (Core Set)
1x The Winds of Winter (Core Set)
1x Threat from the North (Princes of the Sun)
1x Fear of Winter (Defenders of the North)
1x Fury of the Dragon (A Clash of Arms)
1x A Time for Ravens (A Time of Ravens)
1x A Song of Summer (A Time of Ravens)

I appreciate all the advice you've given. I'm not sure if I need all the winter, but the only thing I can see myself replacing it with is more The Hatchling's Feast. All-in-all, unless I get some advice that strongly convinces me that Winter isn't the way to go, I like this deck. Once again, thank you for all the advice and help. Oh, and if my opponent has no blockers, he/she cannot keep me from having unopposed challenges. That is how I shall win.

The problem will be that everyone will probably be running 3x Narrow Escape. If you Westeros Bleeds, they will Narrow Escape and get back all their characters and you will have none left. You don't have enough characters (especially shadows) to recover from something like that. So you will likely have to discard your hand, which will probably be full of nice events that you didn't have enough influence to play. Not really sure how Targ could recover from that, except maybe more character recursion, or maybe Rhaenys' Hill. You'd have to be first player and play Rhaenys' Hill before Narrow Escape after a Valar or Westeros Bleeds...

If I'm playing the deck right then such things shouldn't bother me, besides, I'll probably make them waste such spells while I'm burning their characters away anyway. If that's the only concern at the moment then I still don't see a major problem with the deck.

However, I want to thank you for your time and consideration, and will have to note that as a card to keep in mind as I play.