Best achievment point earners?
getting all slayer achievements, JP, and other general PvE achievements, also crafting achievements perhaps?
Doing Dailies (PVE and PVP) every single day. While doing those, get any other achievements that’s all you need.
Best and fast are not exactly the same actually. Fastest would be to grind the easiest ones: as said above general pve especially explorer and hero, with a bit of PvP (earlier tiers are rather easy). But this way you will end up with only the hardest (longest actually) to be done. The best way for long-term AP gain is, imo, to progress in the longest ones while you do the easiest (weapon master for example).
In any case: do all the daily/monthly/living story achievements
A time machine to get all those daily and monthly points!