Hi., it is "like she do me, like she done me,..." as Copy points out.
Lennon is using two conjugations of "to do" at once,
Standard English: does, did (present, past)
Non Standard: do, done (present, past)
That Non Standard usage is particularly used by some African Americans, and further, a strong subset of that group would be musicians, both jazz and rock, and all the Beatles confirmed their inspirations came largely from American songs.
In the late 30's, a Mae West film was entitled "She Done Him Wrong." (She did something to him that is not right, something bad.)
As isolated phrases, "she does me, she done me" could have a sexual connotation but Lennon starts out saying that this is about love, so this a down low funky way for him to say, "I'd met all these women in my life because I'm John Lennon the rock star, now this woman has come into my life and finally taught me what real love is, she's transformed me in an overwhelming way, and I hope it's as real I as believe it to be....(don't let me down, disappoint me.)
