1.I don't only study a lot but also play a lot
This is OK
2.I not only study a lot but also play a lot.
This is OK
3.Not only I study a lot but also play a lot.
This is wrong. If you begin with 'Not only', you need an inversion - there has to be a verb before the noun 'I'. Because this is a simple tense, you use 'do' as an auxiliary :
'Not only do I study a lot but I also play a lot.'
None of these negative sentences work with 'not only ..but also'. At a stretch, you could say 'I not only don't study a lot but I don't play a lot, either', but it's a little awkward. It might be neater to say something like 'I neither study a lot nor play a lot.'