Muni Ved VyAsa, the (author of MahAbhArata) was also referred as "Krishna DwaipAyana". His birth was before Krishna.
Krishna was 1 of the many names given to VAsudeva (son of Vasudeva), due to his dark skin complexion. This should have been in good spirit and not racial. Probably it was usual for prominent people to have many names in those times.
Interestingly Arjuna apologizes during Gita 11.41-42 for referring "Krishna" to a person, for whom now he was assured to be God. In Gita alone, many of his other names/addressing can be found.
During MahAbhArata times, like VyAsa & VAsudeva, Arjuna & Draupadi were also referred as Krishna & KrishnA respectively due to their dark complexions.
From Shrimada BhAgavatam (thanks to @Vishu)
Your son Kṛṣṇa appears as an incarnation in every millennium. In
the past, He assumed three different colors — white, red and yellow —
and now He has appeared in a blackish color. [In another
Dvāpara-yuga, He appeared (as Lord Rāmacandra) in the color of śuka, a
parrot. All such incarnations have now assembled in Kṛṣṇa.]