Sounds to me like it is possibly one of the springs in the clutch itself.
I have had them come loose and fall down, only to get picked back up and flung up against the bell housing. Sounds kinda like that little kids fake lawn mower..pop,pop,pop...lol.
A clutch job is just basic mechanics skills. Its harder to put back together than to get apart.
You have to take both driveshafts off.
Then support the t-case tail shaft with a jack stand and take the skidplate off.
I always put a small bottle jack under the oil pan at this point, cause you will need it there for support as well as helping with alignment later.
Next you support the trans with your other jack stand, and unbolt and remove the t-case.
Then comes the fun part, disconecting the line for the hydraulic clutch, and unbolting the bell housing from the motor.
After all is loose its just brute force and lots of swear words and it will come out of there.
Get a clutch kit from NAPA or your favorite auto parts store. The kit will come with a clutch disc, pressure plate, throw out bearing, and a pilot shaft bushing.
Take out all of the old parts, replace with the new parts, and reverse the order of the tear down and you have just replaced your clutch.
I have done many, and i just replaced the clutch in my 87 Wrangler this summer while switching to an updated AX15 trans as well. The job took me approximately 6hrs, working alone. So for someone like yourself that has never done it before, figure on atleast a weekend or so.