I don't want to fix my family friend's computer.
Is there a problem? Why wouldn't you fix their computer?
Yes he once insulted me to my face and I can't look at him in the eyes anymore.
Even though your family friend once insulted you, it is important to try to be on better terms if possible. Not trying to help when you have the opportunity to do so is selfish behavior and should be avoided. If you fix your family friend's computer he will likely apologize for mistreating you in the past and your relationship with him will improve.
I don't care. I wouldn't work for someone who insulted me.
It's understandable to still be upset, but for yourself its better to just forgive and forget. You have the option to help and not doing so is just you being selfish. You're better off just playing nice and maybe they'll owe you one in the future.