iMac help

I have a early 2008 intel 24 inch iMac..a few weeks ago the screen started to flicker really badly so I turned it off and let it sit for awhile..a few hours later I turned it on and got the image below and at this point it wouldn't even get past the white screen..so I unplugged it and let it sit for a day or so..when I turned it back on it started up like normal and everything seemed like it was 100%...after a hour or so the screen started to flicker again and the whole thing froze..so now I know it's more or less shot all I want to do is get it to the desktop once more so I can do a time machine backup before I take it in..i let it sit unplugged for about a week and a half and just yesterday I turned it on and after a few tries it went straight to the desktop w.o any issues..but that lasted for 10 mins only cause now all I get is the screen below and it's running really hot..ugh please tell me there is a way I can get one last backup in

 

I read up on that unfortunately all I have is a iPad and a tower pc w. USB only

 

Would a USB to FireWire cable / adapter work for target disk mode???

 

So target disk mode is basically my only option??