As @Joe commented, there could really be lots of reasons for that behavior. It could be a combination of reasons. And in order to handle it the right way (as opposed to achieving a temporary deceiving quiet), the root causes are better be exposed. There are some root causes.
Let's try to narrow it down. Does your child have no discipline whatsoever? You said "His teachers have expressed that he has been violent at school." It sounds like his behavior is bad only at school. Is it only violence? How about paying attention to the teachers? Doing homework? If it's only violence, is it aimed toward specific (irritating?) children or does it occur on specific days of the week? (like longest school days, or the day after he meets his dad because he misses him etc.).
So my first suggestion is try to understand your child. Try your best effort by yourself before any mental health help arrives or not. Don't be dependent on that. Just do it (as you've started by posting here) because you love your child. BTW, you know him best so you can most probably help him best.
(You might already been doing it, but) My second suggestion is talk to him. Or more accurate: make him talk to you. Let him describe his day at school. Let him explain his ups and downs, his violence, the teachers' reactions etc.. Make him comfortable talking to you. Just try to hear him without criticizing his behavior. Since you're his most meaningful adult in his life, you're his solid rock on which he relay upon. If he's criticized at school, then comes back home and get criticized all over again, where would he go? With whom would he share his thoughts and feelings? Open communication and trust are key properties for healthy relationships and parenting. Moreover, if you succeed in that, you might also come to understand your child's behavior (the first suggestion).
Third suggestion, a non-direct suggestion. Don't blame anyone... Not yourself, not the child, not the father, not the school. Don't look for someone to blame and don't jump to conclusions.
BTW, you mentioned that his dad has mental issues "as well". What do you mean by "as well"? In addition to your child? He wasn't diagnosed yet. Let him enjoy being mentally healthy while he can :). Jumping to conclusion might introduce new problems instead of solving the existing ones ("oh! You're just like ---" and stuff like that might cause children actually think that they should start behaving like ---).
Last note, you say that the punishments you used were to no avail. Typically I discourage punishments, certainly at such an age, but even for those that believe in it, in this case, either your child simply isn't afraid of them (not probable for such an age), or he doesn't perceive them good enough, or that the things you use to threat aren't important enough for him.
I am no psychologist or education expert or anything... Just suggestion for your own judgement of course. And I hope it'll help you in your journey.