"I do not watch a lot of movies."
Usually, the topic is dropped unless it's a new topic or omitting it is ambiguous. The topic is often the subject, though this is by no means a rule (the topic can really be any of the nouns in the sentence, subject, direct/indirect object, etc.).
This gets a little confusing with pronouns because people often drop わたし when it's the subject, even in those cases when the subject isn't the topic (same applies with あなた when you're asking a question, but あなた is also usually substituted by the person's name if it's not dropped). That's why you can say 「えいがは見ます」 and have it mean "I watch a movie." In theory, assuming this sentence isn't establishing えいが as the new topic, you could also drop えいが and say 「見ます」 (this might be a bit ambiguous, though. I'm not entirely sure if it would be better to keep the topic dropped and add the subject, or vice versa (i.e., to clear up ambiguity, is 「えいがは見ます」 or 「わたしが見ます」better?) I think that 「わたしが見ます」 puts a little more stress on わたし, so 「えいがは見ます」 might be better in this case).
This is turning into a pretty long rant on Japanese grammar, so I'm going to stop now. 頑張ってください！
This sentence appears to be following rule number one.
"Rules for Japanese Particles Change in Negative Answers
In general, the rules for the Japanese particles change in negative answers are as follow[s]...
- Rule 1: None/を/が/も → は
Meaning: When there is no particle or the particle is either を, が or も, change it to は.
- Rule 2: Other Particles → Particle + は
Meaning: If the particle is neither を, が nor も, add は to the particle. (に → には, と → とは, で → では, へ → へは, etc)"
You can add し (ません) which is form of the verb する to some nouns to help the noun function as a verb. These are called verbal nouns. Keep in mind that some words that are verbs in English are considered nouns in Japanese.
However, 見る is already a verb, so there's no need to add anything to it.