What would I call my friend's personality?

From what you have described I would say he is modest

From the Oxford Dictionary: Modest Unassuming or moderate in the estimation of one’s abilities or achievements

Or that he is "fishing for compliments" Fishing - Try subtly or deviously to elicit a response from someone

I would say he's acting 'coy'. While this word isn't directly tied into the lying element you insinuate, if he's downplaying his abilities to gauge a reaction from others deliberately, this is the word I'd use. Coy people downplay something to appear alluring or to attract attention from others.

This might not strictly meet the definition of what you're looking for, but it's close enough.

If you were satisfied with describing the effect of being a ________-er , you could (humorously) respond that he is "derelict in his friendship duties" or a "shirker" or "malingerer" or (and this is British dialect) a "fainaiguer".

Or, to get more to the point, I think the guy is a self-deprecator, but people don't usually use that noun. It sounds more natural to say, "OK, dude, stop with the self-deprecation and help a brother out."