There is a lot of skills/traits/auxiliaries/... that add to influence per 'influence-earning situation'. Well what are these situations? I imagine winning battles is one. What about just commanding an army and doing nothing? Being a governor? Not doing anything at all, but being a Companion?
Winning battles for generals, building stuff for governors and simply being around for statesmen.
basically doing important stuff so standing around doing nothing won't earn you influence, but winning a battle earns the general influence, Building stuff in cities and towns will earn a governor(who's in charge of that province)influence, and statesmen will also earn influence when just being statesmen.
some extra info
Statesman, are idle characters, meaning they are currently not doing anything. You can appoint them to a position (candidate, still can do nothing), you can appoint them to an army or govern a province. They usually have the least influence.
Candidates (i believe) can be appointed to an honorary position or office, they get an influence bonus and over time they are able to be promoted further for more influence and bonuses.
Governors are characters appointed govern a province. I recommend doing this for family members since they get a lot of influence along with allowing you to select edits for provinces, to get bonuses.