Always use between you and me.
All prepositions, such as between, take pronouns in the objective case (it, her, him, me, them, us, you) not in the nominative case (he, I, it, she, they, we, you) or possessive case (her, hers, his, its, mine, my, our, ours, their, theirs, your, yours).
- Just between you and me, we are having a party for the family.