I was wondering what would happen if :

a) you take control of an opponents character that can be become an agenda

b) You are not running an agenda

c) The character gets killed (whilst you still control him)

Do that character become my agenda?

Assuming this does occur, what about the following

a) I play River Raid and take control of Griff

Response: After a character card is discarded from an opponent's deck, put that card into play under your control. At the end of the phase, return that character to its owner's discard pile.

Griff

If Griff would be killed, instead attach him to your House Card as your only agenda with the text: "If you control fewer attachments than each opponent, unattach Griff and return him to play

If Griff is killed during the same phase, does he stay as my agenda or would he be returned to my opponents discard pile?

If he stays as my agenda and gets unattached/returned to play in a later round, do I keep control of him

Do that character become my agenda?

No. Never. When a card leaves play (and agendas are not "in play"), it always goes to its owner's appropriate out-of-play area. If you take control of an opponent's character agenda, it dies, and neither of you have an agenda, it becomes its owner's agenda. If either one or both of you has an agenda when the character dies, it just goes to the dead pile.

If Griff is killed during the same phase, does he stay as my agenda or would he be returned to my opponents discard pile?

If a character leaves play, all pending lasting effects on it end immediately. Look at it this way: if you used River Raid to take a non -character agenda character from an opponent and it died during the phase, you wouldn't move it from the dead pile to the discard pile at the end of the phase, right?

If he stays as my agenda and gets unattached/returned to play in a later round, do I keep control of him

Moot. Cards always go to their owners when they leave play as described above.