Jumping in and finally giving space to a thread celebrating the Lannisters!
Why play Lannister?
1. Lannister gives you access to great economy. Gold shouldn't be an issue for you, so you can play plots for effects, not economy.
2. Lannister gives you access to solid draw. When you get your board established, Lanniport, Casterly Rock, and Pycelle and get you cards in a hurry.
3. Lannister gives you access to a toolbox. While currently weak in military, Lannister decks possess many tools that allow you to adjust as needed given your opponent.
4. Lannister gives you access to some of the most potent cancel/control cards. Between Treachery, The Things I Do For Love, and Hand's Judgement, you have a lot of ways to disrupt your opponent's game plan.
5. Lannister gives you access to the best 7-coster in the game! (There, I said it) 
While Lannister No Agenda seems to be a popular deck out of the core box, I've used the following control build to good effect. It's less reliant on Tyrion than N/A decks, and opens up several pathways to victory, even if a key character gets plucked by Heads on Spike or blanked by Milk of the Poppy. This build has even given me moderate success against Bara and Targ, by FAR the toughest match-ups faced by the Lions of Casterly Rock!
Total Cards: (62)
Faction:
Lannister
Agenda: (1)
1x Fealty (Core Set)
Plot: (7)
1x A Game of Thrones (Core Set)
1x A Noble Cause (Core Set)
1x Calm Over Westeros (Core Set)
1x Confiscation (Core Set)
2x Filthy Accusations (Core Set)
1x Wildfire Assault (Core Set)
Character: (33)
3x Burned Men (Core Set)
3x Cersei Lannister (Core Set)
3x Gold Cloaks (Core Set)
3x Grand Maester Pycelle (Core Set)
3x Lannisport Merchant (Core Set)
2x Lannisport Moneylender (Core Set)
2x Littlefinger (Core Set)
3x Ser Jaime Lannister (Core Set)
3x The Queen’s Assassin (Core Set)
2x The Tickler (Core Set)
3x Tyrion Lannister (Core Set)
3x Tywin Lannister (Core Set)
Attachment: (7)
3x Milk of the Poppy (Core Set)
2x Seal of the Hand (Core Set)
2x Widow’s Wail (Core Set)
Event: (9)
3x The Hand’s Judgment (Core Set)
3x The Things I Do For Love (Core Set)
3x Treachery (Core Set)
Location: (13)
2x Casterly Rock (Core Set)
3x Lannisport (Core Set)
2x The Kingsroad (Core Set)
3x The Roseroad (Core Set)
3x Western Fiefdom (Core Set)
Strategy: You need to slow the game down and work hard to keep your opponent from drawing cards. Given your economy you should be able to keep up with them re: characters on the board, but if you can establish even moderate draw while denying theirs, their hand will quickly get choked and then you can wildfire for a reset and close things out from there.
Filthy Accusations is a must-have x2. It's a godsend in games vs. renown decks and also serves to buy you time while you fish for the cards you need to establish control of the board.