Birth-wise Krishna was a Vaishya (the 3rd division), though he lived a princely life like a Kshatriya.
Krishna's caste was Yadu / YAdava or Vrushni.
The castes are like family names, which are carried by a person to his son. Probably you meant classes/divisions, which are only 4, based on qualities: i.e. Brahmana, Kshatriya, Vaishya, Shudra.
A son/daughter's birth-wise class/division is considered according to their father's lineage. For daughters, it changes after marriage & for sons it may change based on merits & virtues.
However son always carries his father's caste (viz. surname) throughout his life.
Now Krishna's mother Devaki might [or might not] be of Kshatriya lineage, however his father Vasudeva was from Vaishya (business) category. To be very specific, in such case the division is decided as below:
The son begotten by a Vaisya upon a Kshatriya woman becomes a Vandi or Magadha. The duties assigned to him are eloquent recitations of praise. [this answer]
Here is how Krishna's class can be considered Vaishya:
Vasudeva was a (real or cousin) brother of Nanda-baba (foster father of Krishna), who was the chief among the cowherds. Same thing is true for many Yadu-s. Moreover it was not unusual in many regions to have a Brahmana or Vaishya or Shudra kings as well. Vasudeva's father Sura also could have been a Vaishya king.
Vasudeva was yadhava king. He was the brother of cowherds tribe chieftain Nanda Baba who was a Surasena. [Wikipedia]
Dhritarashtra said, 'Hear, O Sanjaya, the celestial feats of Vasudeva, feats that Govinda achieved and the like of which no other person hath ever been able to achieve. Whilst being brought up, O Sanjaya, in the family of the cowherd (Nanda), that high-souled one, while yet a boy, made the might of his arms known to the three worlds. [Dronabhisheka Parva]
Another strong reference, is that Kunti was a "real sister" of Vasudeva from same father Sura (or Surasena). Now MAdari refers Kunti as of lower division based on birth.
There was amongst the Yadavas a chief named Sura. He was the father of Vasudeva. And he had a daughter called Pritha, ... gave from friendship this his firstborn daughter unto his childless cousin and friend, the illustrious Kuntibhoja--the son of his paternal aunt [Sambhava Parva]
... [Madari,] the daughter of the king of the Madras privately addressed Pandu, saying, 'O slayer of foes, I have no complaint even if thou beest unpropitious to me. I have, O sinless one, also no complaint that though by birth I am superior to Kunti yet I am inferior to her in station. [Sambhava Parva]
When a Kshatriya begets a child with a Vaishya woman, the Kshatriya order is usually maintained for that child. However the son born under such circumstances, is also referred as "Karna". Kunti's son from Sun before marriage, was actually named as "RAdheya", was also later known as "Karna". It's quite likely that Karna would have been named/referred so, by some intelligent people (like Bhishma, Vidura), finding similarity of his face with Kunti. This further solidifies that Kunti was from Vaishya background, so was her brother Vasudeva, Krishna's father.
During that year, O king, was begotten upon her by the illustrious Dhritarashtra a son endued with great intelligence who was afterwards named Yuvutsu. And because he was begotten by a Kshatriya upon a Vaisya woman, he came to be called Karna. [Sambhava Parva]
Even the Krishna's Narayani army, consisting mainly of Yadava-s, was referred as the army of cowherds. The profession of cowherd is considered for Vaishya division as per BG 18.44.
In the left foot, O monarch, was stationed Kritavarma accompanied by the Narayana troops, and those invincible warriors, the gopalas [source]
Usually a person with non-Kshatriya quality used to take the top ministry under the Kshatriya king, e.g. Vidura, who was born to a Shudra maid or Yuyutsu, born to a Vaishya queen. Similarly Vasudeva was also a prime minister under Kamsa's kingship. Moreover many times Krishna has been referred belonging to cowherd tribe, even though his time with his foster father Nandababa was reasonably less:
Duryodhana began to afflict Vasudeva with keen and bitter words, "O son of Kansa's slave, [Shalya Parva]
Bhurisravas said, ... thou hast, unseen by me, cut off my arm ... A man quickly catches the behaviour of those with whom and amongst whom he moves. This is seen in thee, O Partha! Being of royal lineage especially, in Kuru's race, how hast thou fallen off from the duties of a Kshatriya, ... Who else, unless he were a friend of Krishna's, would inflict such a wrong ... The Vrishnis and the Andhakas are bad Kshatriyas, ever engaged in sinful deeds, and are, by nature, addicted to disreputable behaviour [Jayadratha-Vadha Parva]
Kansa: I will put to death Vasudeva himself, the cowherd Nanda, and my foolish father, Ugrasena, [Vishnu Purana]
An indirect reference: Krishna's one of the name "Rannchhod" is due to him running away from the battlefield. This is least likely to be a Kshatriya like quality as per Gita:
BG 18.43 - The natural duties of the Kshatriyas are heroism, boldness, fortitude, capability, and also not retreating from battle, generosity and lordliness.
Traditionally the "cowherds" community belonged to the "Vaishya" (business) class. Refer BG 18.44. Now there are few references, which again suggests Krishna's lineage towards that:
the king of Chedi (Shishupala) could bear no more, He then replied in rage unto Bhishma in these words... O Bhishma, in unceasingly praising with devotion, from ignorance alone, Kesava so unworthy of praise. How dost thou, from thy wish alone, establish the whole universe in the servitor and cowherd of Bhoja (Kansa)? [Sisupala-badha Parva]
Of course being a living symbol of supreme reality, the divisions don't bind to the lord Krishna. However, I am just being in the scope of the Qn and discussing Krishna's "birth based division".