More than likely, like habanero said, it's time for some carb diaphrams.
Like instances or problems also happen when someone cleans the old diaphrams with carb clean which coats and stiffens the diaphrams. Cranks and will run on a slight partial choke, but nothing else... Sound anything like what yours may be doing?
Still could be your fuel lines cracked and busted... ANY 3/16 fuel line from any auto parts store should work. It's a good thick rubber meant to be used with gasoline, plus it fills out the hole that the fuel line passes through going from the tank to the carb the best "helps keep trash out".
Carb kits are still fairly easily obtainable for these saws. Kit number is RK-88-HL and will run right in the neighborhood of $15 - $20 from your local small engine shop. This is the "major" rebuild kit. I don't know if the regular diaphram kit is still available or not, but still see them for around $7 on ebay quite a bit.
Also other items still available for these saws are the piston rings from jackssmallengines.com (#55038 for C5, C5G, C51, C51G, C52, C7, C7G, C71, C71G and C72, then #55537 for the C9, C9G, C91 and C91G). All of your crankshaft bearings "other than connecting rod needles" are available through MotionIndustries.com (flywheel side ball bearing #6203ZZNR "will need to remove one of the covers on the bearing - it's a completely sealed bearing and only needs to be half sealed" and #B-1210 needle bearing for the clutch side). You can also get the ball bearing "a little better quality" from Napa, but cost $10 more #6203-2ZNRJ and it will also need one of the sealed sides removed. Crank seals on the other hand aren't that common, atleast the flywheel side, because you have to use NOS #58308-A, BUT the clutch side for the C5, C7, C9 is still easily available from MotionIndustries or Napa #7414.
If you don't mind paying $150 you can have your cylinder rechromed through usnicom.com...
Other seals are still obtainable as NOS through ebay occasionally, but I just make my own from templates I've made of all of the seals on the C-Series saws. For the gasket material I use felpro's rubberized gasket material... I find it seals better than the regular paper junk, plus it's not too thick like some of the materials out there.
Also if you don't know of it yet, a great saw specs website is
www.acresinternet.com/cscc.nsf ---
Hope this is all of more assistance.
Greg