What is considered a "weak key" in AES? I need to construct an AES key from an array of bytes (in Java), but I first I have to check if the key created from these bytes would be weak.

Java can check for weakness for DES, but not for AES. What are the criteria for an AES key to be considered weak?
The comments already have covered the two main points, but let me try to put it in the form of an answer. There are not (that we know) weak keys in AES, in the sense that you cannot formulate a routine $isWeak(key)$. However, there are weak ways to generate an AES key (i.e. bad randomness). 

An AES key is just a bit string of length $n$. That means that there are $2^n$ possible keys. If the key is generated by a properly seeded CSPRNG, AES is (computationally) secure. If you don't know about the origins of the key, or it is generated by a less secure RNG (such as Java's Math.random), the key shouldn't be used.