We only get what Alex hands us, so we've got to be mindful of both the perspective and biases inherent to a first-person narrative.
The advantage to this is that we get extremely intimate with and engaged in Alex's life. After all, it's an "insider's view" we're seeing, albeit from only one lens. Despite all of the senseless brutality he inflicts upon others, for example, we come to like and forgive Alex, because we're stuck inside his head.
The disadvantage is, well—we have to spend a whole lot of time with Alex.
Ah, just kidding: Alex is an excellent (if thoroughly unlikable) character. The problem with the first-person narration is that we aren't privy to how others view our protagonist-narrator (except when it's obvious – like when P. R. Deltoid spits on Alex's face), and so we can't be exactly fair or just in our assessment of each situation.