Conflict between player-characters is a double-edged sword. Depending on how the players deal with it and how well the campaign and game system supports it, it can either add a very interesting aspect to the game or ruin it completely.
Which level of conflict you want between the player-characters is something the group should agree on before the campaign. Possible variants are:
- Player-characters must be nice to each other at all times and act as a team as well as they can.
- Player-characters can have verbal conflicts and ideological disagreements for show, but still need to act in the interest of the team.
- Player-characters follow their own agendas and may put them over those of the party (just make sure you don't let them get in the way of a good story). PC-on-PC violence might happen, but should be very well-justified in-character and out-of-character and rather the exception than the norm.
- Player-characters are direct opponents. Defeating each other is the goal of the game.
Apparently you are further up in that list, while the other player is further down. When there is a misunderstanding about which of these variants is expected among the group members, it can negatively affect the game experience of everyone. That's why it's important to agree on it and then stick to it. Also, the GM must be aware of the tone and plan accordingly. When the PCs might turn on each other at any time and the GM doesn't have a contingency plan for this situation, the whole campaign might fall apart.
Regarding your current campaign: If your group agrees on a cooperative playstyle from now on, and the personality of some of your characters are simply not designed for that, there are two options your group (including GM) can agree on:
- Change the characters to have more compatible personalities. If necessary, decide retroactively that certain problematic events have never happened. Yes, you can do that. It's your game.
- In agreement with the players remove the problematic character(s) from your party, then replace them with new characters. Kicking them out actually is an option. When everyone hates the guy, he might even return as an NPC villain, which could turn out as a really awesome RP moment (maybe he was a mole all along?). Or just have them die.
But whatever course of action you take, get buy-in from the player first. Just removing their character from the game without their consent and handing them a new sheet to fill out won't be a very nice and mature way to handle the situation.
