Overtaking a car before and he sped up and tried to not let me past

I don't agree with both of the actions.
Yes, you were wrong to have gone up his backside to intimidate him to go faster. I tend to stick to 30mph if a road is 30, and I stick to 40, if a road is 40 most of the time. That is something that annoys drivers, me included, to look in your rear view mirror to see a car revving etc to intimidate you.
So that is point 1.

Point 2, as you went to overtake, he shouldn't have sped up. That is him in the wrong.

As the driver of the car infront of you, I wouldn't have wanted you to have been up my backside any longer, so I would have let you overtake.

This. Rather than speed up he should have just let you pass. People should just stick to their own driving. If the OP wants to take a risk by going to fast its their problem.

Exactly. I have been in a situation kind of like this before.

A bus was parked up at a station, and as I passed, I was like a 1/4 past it, it pulled out on me trying to enter the lane I was in. Luckily no cars were on the right hand lane, and by this point, I was already doing around 30-40 so thought it best to speed up quickly and overtake, than quickly slow down to jump back into the correct lane. The bus if it had sped up, would have put me in even more danger than I already was.
I quickly sped up, overtook, and went back into the lane I was meant to be in.

Right, so the guy (30mph guy) thought 'I don't want him to overtake me but instead, want him to remain dangerously behind me'. He was being a ****.

Why was he overtaking? You and him are an idiots. There is a speed limit for a reason. He was going at the speed limit, you tried to make a dangerous by overtaking.

I was coming home before to pick someone up to take them the hospital as there was a minor emergency, I was driving behind a car which was driving at about the speed limit but I was driving a little to close to him as I was in quite a rush, I could see he was looking at me with anticipation through his rear mirror as if he knew I planned to over take. I stayed behind him as although there was chance to overtake I was approaching arounabout.

After the roundabout there was a long 40 MPH strait and no cars were coming in the opposite direction, I decided now was a good time to overtake so I got my speed to 50 MPH and began my overtake, Then the driver of the black car sped up and before I knew it I was doing 60 mph, I then saw another car approaching so I had to get back in my lane, yet the black car wouldn't let me past.

I slowed right down to about 30 MPH and got back behind the black car, then he slowed down again below the limit in front of me!

Why did he do this, it really pissed me off and put me in danger!?

I was coming home before to pick someone up to take them the hospital as there was a minor emergency, I was driving behind a car which was driving at about the speed limit but I was driving a little to close to him as I was in quite a rush, I could see he was looking at me with anticipation through his rear mirror as if he knew I planned to over take. I stayed behind him as although there was chance to overtake I was approaching arounabout.

After the roundabout there was a long 40 MPH strait and no cars were coming in the opposite direction, I decided now was a good time to overtake so I got my speed to 50 MPH and began my overtake, Then the driver of the black car sped up and before I knew it I was doing 60 mph, I then saw another car approaching so I had to get back in my lane, yet the black car wouldn't let me past.

I slowed right down to about 30 MPH and got back behind the black car, then he slowed down again below the limit in front of me!

Why did he do this, it really pissed me off and put me in danger!?

I was coming home before to pick someone up to take them the hospital as there was a minor emergency, I was driving behind a car which was driving at about the speed limit but I was driving a little to close to him as I was in quite a rush, I could see he was looking at me with anticipation through his rear mirror as if he knew I planned to over take. I stayed behind him as although there was chance to overtake I was approaching arounabout.

After the roundabout there was a long 40 MPH strait and no cars were coming in the opposite direction, I decided now was a good time to overtake so I got my speed to 50 MPH and began my overtake, Then the driver of the black car sped up and before I knew it I was doing 60 mph, I then saw another car approaching so I had to get back in my lane, yet the black car wouldn't let me past.

I slowed right down to about 30 MPH and got back behind the black car, then he slowed down again below the limit in front of me!

Why did he do this, it really pissed me off and put me in danger!?

So let me get this straight you were trying to overtake someone who was doing the speed limit in a 40mph zone and so you tailgated him, well sorry but you should not tailgate someone because if he put his brakes on or someone or something ran out in front of him then you would have gone into the back of him and therefore it would have been completely your fault. Yeah he was stupid as well for suddenly speeding up, but you were being just as dangerous.

The OP says that he was driving a little too close to him, that is not the same as tailgating.

Also, how do you know what he judges as a little too close? According to the facts that you learn before you do your driving test, 35m would be a little too close at 40mph.

Lastly, let's say that he was driving too close to the back of the guys car. The guy slams on his brakes, what's going to happen? He's going to stop and get a shunt from the rear, it won't do him any harm, sure it'll damage his car and will be an unnecessary accident, but nothing major at all.
However, by accelerating to prevent somebody overtaking you you put them in a situation where you are forcing them to remain on the wrong side of the road, travelling towards any oncoming traffic that may be on the road at the time, making a head on collision at speeds which could easily be fatal.

How exactly is endangering somebody's life, just because they have annoyed you a bit, acceptable?

If somebody is tailgating you then you either slow down so that they can overtake, thus removing you from the dangerous situation, or if you are in a place where they can't overtake then you brake check them (brake hard for a fraction of a second, so it looks like you're slowing down a lot but in reality you only drop a couple of mph, because often the shock will cause the driver to back off). You should never take a course of action which turns a very slightly dangerous situation and turn it into one with a potentially fatal outcome.

Well the OP has already suggested himself he was driving too close to the car in front which is dangerous especially if they have to slam the brakes on and you hit someone at 40mph, it's gonna hurt and do damage. I'm just saying a 40mph zone isn't a good place to overtake anyway, yeah the other driver was foolish for speeding up as well. But both situations could have resulted in an accident either way.

But unless you're driving literally on the bumper of the car infront then you're not going to hit them at 40mph. You are more likely to hit them if you're driving closer, but you're still going to have time to react and hit your brakes first, so you'll be going slower than the speed you were previously travelling at, combined with the fact that you don't know anything about the road surface, the cars involved, the condition of their tyres, the reactions of the drivers, all of these things have a huge impact on what is a "safe" distance to leave between cars.

As for "both situations could have resulted in an accident either way", yeah, they can, but the difference is that one would be an accident where you ended up with a pissed off driver with a damaged car, and the other is one where you could very easily end up with one dead driver.

I think we will have to agree to disagree

So you view a fender bender and a fatal crash as equal?

You would be equally upset whether you're parents were involved in a shunt or were killed in a head on collision?