Does it count as "cloud mining" when you mine on your own computer?
Or is that if you mine on some service like this?
https://www.digitalocean.com/?refcode=35eb2c34ba3aAlso if you have managed to get everything up and running on your own computer, how hard is it for a noob to get it up and working on a service like the above?
No its just if you do on a service like Digital ocean, you can use windows azure too.
Its not hard to do, just follow the steps described in the first page and you will mining on cloud.
Good luck

I looked at the steps in the first page but they seem so complicated? I get lost at step 8-9 already...
On my computer i just needed to download the zip from 1GH and then adjust the _start.bat file with my Wallet adress.
Cant I do the same from a cloud computer that I hire on digitalocean? DO i need to go through all those steps, i dont understand them :/ Can please explain?

i supose that till step 8 you´re done.
Once you receive the user and password at yoour email, log in to your droplet via SSH (PUTTY) You can download for free. Google it.
The follow the steps. You have to tipe or copy/paste the code once per line:
8. Relocate into the home directory
Code:
cd ~
9. sudo dd if=/dev/zero of=/swapfile bs=64M count=16 press intro and wait till its completed then tipe the next command from the 9 step.
When you finish all, change the wallet adress posted in the command line to run the miner for your adress.
lets see..
I opened Putty and put in my IP adress I got in my email.