Band of Bastards
A 1st-level character begins the game with 5 Action Points. For every level after that, your character gains 5 + 1/2 your new level. If you create a higher-level starting character, you get the aforementioned AP for whatever level you’re starting as, plus 1 AP per previous level. You also earn 1 AP per 1000 XP, which you receive immediately upon earning the XP.
For the Band of Bastards campaign, we’re starting at 3rd-level, meaning everyone begins the game with 11 AP (5 + 1/2 of 3 + 1 per previous level + 1 per 1000 XP).
You may also gain Action Points whenever you roll doubles on an attack roll, to be spent immediately as part of the attack action. You could add damage, take a secondary attack (like a headbutt), emulate a fitting feat, perform a special attack like a disarm, etc. The use of this impromptu AP should be cinematic and further the game, not just give another boring attack.