things to consider:
If he is using facebook, does he have access to the email as well?
how strong was the password he had in place?
Does your friend log on from home or at work as well?
How long has it been compromised?
How is the grammar in the messages?
Send him a message on facebook, letting him know that you are going to pursue this if he doesnt desist. It may be a prank, in which case it would be bad to have a friend arrested.
while his computer may be infected, realistically, the only kind of person that would sent hate mails, etc. would be a script kiddie(which would not infect from a website most likely) or someone your friend has made angry.
Not to contradict ACG, but if it was a infection due to a popup, etc, the amount of computers infected would be so massive that to have a blackhat(most likely chinese) send out hate mail, etc would be so against the odds, that it can usually be ruled out.
Most likely the infection was due to a keylogger in place, either hardware or software.
In which case everything typed would be compromised.
I would look for anything plugged into the keyboard port on the tower if he has one. Otherwise check the process' and run a hijack this.
facebook has measures in place to resist brute force attacks, so unless he used spoofing, etc. etc. that wouldnt have worked, in which case if he used advanced techniques you prolly cant track him down anyway(on account of proxys, etc)
As has been said before, there are a number of ways to get a IP, but to actually translate that into anything more than a town location, requires a court order. Unless you compare it with known emails on your own.
Remember, if you do by the grace of god, track him down...do not do anything offensive to him, if this was to ever to go to court(like if he siphons money, exposes trade secrets, etc) then you will also in turn be prosecuted.
Also facebook more than likely will not be any help(except perhaps changing passwords, etc), while I am sure they log IP addys, they more than likely wont release, like most companies wont...without a court order.
Remember anything you do will be null if a keylogger remains on the computer or if it is reintroduced again.
I guess my point is, that he most likely knows the person, he may work with them, etc.
If you have any questions feel free to message me, Ill try to respond as quick as I can.