Don't know how big a place you'll need, but finding a 3BR+ rental in the better areas (mentioned by others above) might be a bit of a stretch within the budget you've mentioned.
During the workweek, I'm staying in a furnished rental in Hamilton East very close to the University. It gets a bit loud on the weekends, but since I'm from NYC I don't find it particularly objectionable.
Even in the "bad" areas mentioned above, BTW, I wouldn't hesitate for a second to walk around alone at night. I guess it's all perspective, but for me there are the "bad neighborhoods" that I knew back in the US, and what pass for "bad neighborhoods" in NZ. Simply no comparison.
Oh, and I second Kim39 regarding surrounding towns. Most of my work colleagues don't actually live in Hamilton -- Cambridge, Te Awamutu, and Raglan are popular (although Raglan is a +40-minute drive).