In a software company, how one should ask for designation upgrade. I am working as SSE but doing the role as lead. So I would like to ask for on paper role as team lead. Should I talk to my manager or HR?
To be given a "designation" or title usually requires manager approval, as your manager knows best your duties and qualifications.
That said, how do you know you are doing the job of "lead"? Is there a job description? Are you following it?
Also, has you manager asked you to perform this role? Sometimes, either by chance, by need or out of eagerness, a senior or other person will "step into" a lead role. Usually a discussion with the manager should clear things up.
HR should be there to make sure there are not issues like discrimination, pay grade discrepancies, proper paperwork filed, etc. (For example, sometimes a lead is required to have certifications, sign NDA's, etc.)
If you start with HR they will only ask your manager for the detail relating to your request. In my experience of organisations, managers will make a recommendation to HR to make these sorts of changes.
If you are not asking for a pay rise and you are able to demonstrate to your manager that you are performing a team lead role then it could be straight-forward. If you are asking for a pay rise then I would suggest a different approach.