I have a character that can do them all but requires a bit of a respec for Simulacrum. I like to farm the splinters myself rather than buying them.
Arbiter: I'm not the best and I do die occasionally, never tried uber. I do die occasionally in citadels and they don't spawn much for me, so this is the one I've killed the least.
Xesht: with the right combo of tablets I can do a Xesht every 1 to 3 maps. I like farming it but I haven't been terribly lucky with his drops.
Olroth: takes me longer to get the splinters and has a bit disappointing loot.
Simulacrum: my favorite but takes me a long time to get the splinters and requires a respec (I need 80% pierce chance because Palm of The Dreamer's POPS don't work). The key for making profit is to be able to clear the last 3 waves, they drop considerably better rewards.
King of Mist: I hate the wisps-chasing phase. Ritual is my main this league so I've done heaps and the loot has been disappointing (except for omens).
Kulemak: not enough invitations for me, the unique drops have a really low chance to be crafted nicely.
Atziri: always play safe and sell the unique regalia unidentified, trust me. I like to have fun crafting rare regalias with a random "allocates passive nodes" (never profitable though). I do not like running the temples before (I don't do snakes).
Trialmaster: I hate the elevators, slow-opening pillar doors and red-on-red mechanics. The reason the fragments are so expensive is that lots of players despise the mechanic.
Zarokh: should be your focus if you have an average build and want to build wealth. The last flame = guaranteed 3 mirrors, if you make it your league goal you are going to get one eventually (or you are going insane before then if you don't enjoy sekhema's trial).