My school (which was super snooty) took donations of paintings of people who thought of themselves as notable alumni. Most were usually just filling space, but the more notable you were the more you moved up in spots (from the back ends of the school and common rooms, to the dining hall and main hall etc) the place was coming down with paintings of bishops and lawyers! I imagine Hogwarts snobby-ier alumni did the same - once you got head of department or society success you donated a portrait just in case you made a big enough splash to be of note.

I had a character AGES ago inspired by one of Kaytes wanted's whose portrait was a fairly good personality match for her because she had had it painted when she was a debutante and had kept it into her 70's. My understanding is that the longer the portrait is with the subject the more it comes to be like them, as well as the extent/skill of the magic involved.

I can't remember if it's canon, or maybe some fanfiction I read, but that Hogwarts headmasters have their portrait done immediately upon starting the job and it hangs out in their private office, soaking up their personality until they leave the post by retiring or die, which is usually years and years, which is why they are better than the average portrait.