Why is this phrase considered to be a phrase containing a regular verb? "John has become a rather good actor."

This is not a phrase; it's a sentence.

I see one of English's IRREGULAR verbs: "become."
http://grammar.ccc.commnet.edu/grammar/verbs.htm#irregular

The phrase "John has become a rather good actor" is considered to contain a regular verb because the main verb in the sentence, "become," follows the typical pattern for regular verbs in English.

Regular verbs in English follow a consistent conjugation pattern in the simple present tense, simple past tense, and past participle form. They usually add "-ed" or "-d" to form the simple past tense and past participle. For example, with the verb "work," the simple past tense would be "worked," and the past participle would also be "worked."

In the sentence "John has become a rather good actor," the verb "become" is used in the present perfect tense, which is formed by combining the auxiliary verb "has" with the past participle form of the main verb. The past participle of "become" is also "become," which follows the regular verb conjugation pattern.

To determine if a verb is regular or irregular, you can refer to a dictionary or verb conjugation charts, which list the various forms of verbs. In the case of "become," you will find that its conjugation follows the regular verb pattern, confirming it as a regular verb.