I was going to suggest a similar thing to Runner1409 but put those bits out of the PC into a back pack (wrapped carefully obviously) - if you have it on your back more often than not they will NOT weigh it at your end and no probs bringing it into Qatar - you will have to go through the usual airport scanners but no worries about that! It would be much safer to carry it onboard.
You can get a case, keyboard and mouse very easily here and monitors (good ones) are not that expensive. I just bought an LG 23" LED monitor and it cost 735QR. Don't know what your currency is but in Aussie $, that it pretty cheap! :)
At worst just bring your hard drive and get the rest here. There are many shops that sell computer parts here which would be sufficient for your use I am sure!
Good luck and WELCOME to Doha!