The choice is yours!
It sounds odd to be "talking about movies" and then ask what a person likes to do/doing?
The question is being addressed to a familiar person (second person singular, familiar form of you, or to a young person for example).
Some verbs and expressions are often followed by "-ing" : enjoy, hate, love, like, dislike, (not) mind, can't stand, can't bear.
After "love" and "can't bear" you can use "-ing" or "to + infinitive".
With the verb "like", it does not really matter whether you say "I like doing" or "I like to do".
However, we usually say "I like doing" when "like" means "enjoy" : I like cooking (i.e., I ejoy it).
When "like" does not mean "enjoy", more often than not, we use "I like to do" : I like to do something (i.e., I find it good or right to do something). Example : I like to wash my hair once a week. (It doesn't mean that I like or dislike it particularly, it means that I think it is the right thing to so.
So, depending on what your sentence means, and with the added thing of the interrogative form (use of "do" and inversion of verb and subject), you have a choice.
English Grammar in Use, Teacher's Book, Cambridge University Press