I think I have a sort of loose understanding of how the words differ and what they mean, but I'm really not too sure. I've checked out weblio and goo and ALC to look around a bit first. Can anyone help clear it up?
人情【にんじょう】 is a common (and maybe a bit nostalgic) word which refers to (good) human emotion. You can basically think 人情 includes 友情 (friendship), 愛情 (love), and compassion toward others in general.
人性【じんせい】 is very rare at least as a Japanese word, and I don't know how to use this term. Google gives Chinese articles about 人性 in the first page, although I live in Japan. Perhaps you can ignore this word, but after looking at a dictionary, it seems it's more like 人間性 (human nature, humanity).
人情 is affection or kindness while 人性 is human nature that's relatively used in negative connotation.