I am applying for PhD programmes very soon and I have recently been trying to get in touch with my potential referees. At my previous institution there is a professor who knows me the most out of all academics I know; we had plenty of conversations regarding the field I want to pursue my PhD in and also I did very well in modules taught by him. I was planning to request a reference from him and I tried to get in touch with him three times (through e-mail), however, he has not replied to me (it has been about two months since my first e-mail). Should I assume that he overlooked my e-mails (he is very busy) and send him another message? Or rather, should I try to contact him through the Department? It might seem to be a trivial case, but I really do not want to annoy him too much as his reference would potentially be of the biggest importance in my applications. What would you suggest in this particular case?
I had a similar experience with a professor while working off an Incomplete. He did a lot of outside consulting on top of his academic obligations and was one of those people who is simply hard to contact.
In the event that your professor is simply hard to contact I would track down a phone number and try a direct call. It is possible that your emails got lost in the shuffle or the spam filter.
Alternatively their situation may have changed since you were there: they could be on sabbatical, sick or they may have left the institution. Do you have any friends still there you could check with? You could also check the course listings to see if they are currently teaching.
Without knowing the professor I think it is unlikely they are trying to duck your request. If they didn't want to recommend you, there are other ways to handle that such as with a weak recommendation or simply telling you.
I would try to exhaust the other contact options before officially trying to contact them through the department.
Does your new potential school have a portal for recommendation letters? I know several schools that request the recommendation directly from individuals you specify.
If he has a secretary, call him or her, and ask what would be the best approach.
You could try calling him during his office hours.
If you want to leave a voicemail and are concerned he might be working in his office in the evening -- a safe bet would probably be early morning.
There are scores of questions on this site about professors not sending in a promised recommendation letter on time. Since this guy is even flakier than that, I suggest you look for other professors to recommend you. If this guy manages to send a letter in, you'll be pleasantly surprised. But if he doesn't, your application won't be held up.
A bird in the hand is worth two in the bush.