To my (British) ears, this sentence with the contraction 'it's' sounds perfectly fine. It could stand for 'it has nothing to do with me' or 'it is nothing to do with me'. It's possible that the second form is heard more in British English than American English.

You'd have to ask the author to be sure. It could be either but "it's is more likely. In the first place, it's not very common to contract "it has" that way in the meaning of possession. (It's more common when used as an auxiliary verb.) Also, there are two expressions that overlap in meaning: "to have nothing to do with", and "to be nothing to do with". It's not a clear-cut division, but I would say the first tends to mean to reject or to spurn, whereas the second tends to be used with the meaning of not having any relationship. Presumably it's the second meaning that it is intended here.

it has

Delete it's, insert it has. That will make it correct.