You accomplish a task or duty. It is synonymous with fulfil. You don't accomplish to do or do something. Accomplish plus -ing is unusual. Perhaps you could use it if the task was a gerund. But I would try to avoid this kind of collocation. You may have translated from a Spanish word which can be used like e.g. manage to do. If you are thinking of a verb like conseguir, then its usage corresponds closely with "manage + to infinitive", not "accomplish".

I would say " accomplish doing''. (''accomplish to do" is not correct)

Accomplish doing

Thank you so much for replying. I appreciate your help!! Thanks!!

The simple answer - as Michael has explained - is that neither of these is correct! We don't follow 'accomplish' with a verb at all. 'Accomplish' is followed by a noun - you accomplish SOMETHING, such as a task. Possible expression with verbs are 'manage to do' or 'succeed in doing'.