Answer:
Become is an irregular verb. Became is the past form of become.
become = present
became = past
become = past participle
John has become a rather good actor. This is present perfect. Here the verb become is in it's past participle form.
John has became a rather good actor. This is not correct. Don't use past form with has/have. To make a past simple sentence just use the past form of become:
John became a rather good actor.