A gerund is a non-finite verb form which functions as a noun, it can be modified by an adverb, and it can also take a direct object. The "-ing form" is used in English to indicate to the gerund specifically. This is the case of "becoming" in your phrase.
The gerund phrase in this sentence would be "becoming a movie star" because of the verb "becoming" which acts as somewhat of an introduction to the rest of the phrase.
