Remember the tenets of the Oath of Vengeance:
Fight the Greater Evil. Faced with a choice of fighting my sworn foes or combating a lesser evil, I choose the greater evil.
No Mercy for the Wicked. Ordinary foes might win my mercy, but my sworn enemies do not.
By Any Means Necessary. My qualms can’t get in the way of exterminating my foes.
Restitution. If my foes wreak ruin on the world, it is because I failed to stop them. I must help those harmed by their misdeeds.
So the first thing to do is figure out what exactly you are primarily avenging? Who pissed you off to such a point that you go all-in to take those foes out? And when you see them-- no mercy, no retreat.
This may quite often put you in direct conflict with other members of the party if your sworn foe is not an "always evil" kind of creature or force. Others may want to interrogate these foes, but your Oath makes you swear to just eliminate them from the face of the planet. Or others may want to careful strategize how to go after them, whereas you will just dive in headfirst to go after them.
Once you decide who it is that your oath is against... figure out your character's attitude and emotional state / stability when in their presence. How do you act? Are you rational or irrational? What happens when others (or your allies) get in your way? Would you attack a friend who tried to stop you from going after a sworn foe? Once you make some of these choices, it will make playing the character easier.
I myself am playing an Oath of Vengeance paladin in my Pathfinder game right now, and his oath is the elimination of all demon-kind. The group knows I will rarely if ever negotiate with demons and thus they accept the situation and just let 'Andre be Andre' in those situations. As I am the tank of the group, they know that I am usually only putting myself mainly at risk, so they don't get in my way. The one time when one of the other party members did have to make a deal with a demon in order to remove a curse they had on them, I basically voluntarily made Will saving throws to see if I went after the demon regardless of the fact the party member needed its help. The demon did remove the curse for my ally, and at that point the others in the group basically distracted me so that the demon could make his escape. There was a bunch of roleplay after that when I was pissed for a little while, but they successfully made the case that all of us as a group were geared towards taking on a larger demon army so why give that up over one demon? And a Will save later, cooler heads prevailed.