I would use a good outdoor paint. Maybe oil based. Sand the old paint off as best as you can and if need be, prime it and then paint. Make sure you pick pretty color so you don't embarrass your chickens!
not really referring to the outside here, but after cleaning out the inside really well, make a whitewash with DE (diamacteious earth) (ya i didnt even spell that one right!). Paint the inside of the doghouse and let it dry for about 24 hrs before inserting chickens.
It will kill bugs left over and the previous owner and prevent new bugs from growing/living in there. And then do it twice a year when you normally clean the whole thing out to sanitize it.
We are working on coop #2 and will be painting the inside of ours. We didn't do the first one, mainly because we didn't know what DE was, so we figured when we do our yearly cleaning, we can add it then.
*Don't* use oil paint as it is more likely to need touch ups and repainting than latex. Oil is fine for some things--even ideal is some situations--but I wouldn't use it for exterior wood. The damp gets down in tiny little cracks in the paint, then can't get back out through the paint and causes wood rotting and the paint to separate from the wood and pop off. Otherwise, yes, get a good quality paint and go for it.