I did my graduation from HPU with 70% marks but didn't qualify entrance test, so best private university or college for M.Sc chemistry?

You can apply for M.Sc in chemistry in Chandigarh University.

There are many private universities in Southern India where you can take an admission but, a good preference can be Loyola College, Chennai. Further details are here: https://www.shiksha.com/science/articles/top-10-bsc-msc-colleges-in-india-blogId-13473.