Sounds like a good starting point for Grim. Feel free to message me privately some backstory whenever you get the chance.Currently brainstorming a bit on the following player character:
race: Half-Orc OR Orc (I have Volo's Guide to Monsters, so I could use that version)
class: Rogue (Arcane Trickster) OR Wizard (Enchantment)
alignment: Chaotic Evil (not the extreme sociopath/psychopath type, she is Chaotic because she has a strong disdain and resentment against any authority, and she is Evil because she believes that the ends justify the means)
faction: none (clueless)
Short background: Grim has been in conflict with orcish culture, rules and authority since she can remember, until she was finally banished by her tribe after a serious act* of sedition and defiance, and is now on her own lonely path to leave the whole life on the material plane behind her back
*to discuss with the DM about how much of said act will be public and how much will part of the 'dark secret'
I am not at all opposed to the PCs being given secret and possibly conflicting purposes but the DM must guarantee that this cannot and will not cause PvP conflict to the extent of mechanically-resolved physical confrontation or betraying the party to their doom.
Some examples to clarify:
Say that PC1 and PC2 have conflicting personal goals, and PC1 ruins PC2's personal goal. An argument starts, and PC1 and PC2 beat the sh*t out of each other in a fight, that the DM resolves through narrative with the consent of both, and ends up effectively without mechanical consequences for either, and a resolution agreed together. So far this is OK for me, just part of the narrative.
It is not ok, if the DM has PC1 and PC2 actually resolve the argument by rolling dice against each other or even a full combat by the rules, or if the DM allows either character to take some hostile action such as casting an offensive spell or stealing equipment from the other, whether this is resolved by the rules or DM's fiat.
As to betraying the party as a whole, that's a bit more complex... I think it's best avoided, but I can see some possible appeal in the event that the player of such character has actually decided to leave the game, and the DM has come up with a plot that effectively turns the PC into an NPC. In such case, the betraying character "loses the game" (in the sense that he won't be playing anymore), and the rest of the party continues to play. What shouldn't happen, is that the betrayal results in a TPK or such other extreme outcome that causes the other players to quit the game as well.