M10 to remove extractor...
The tool you're talking about is the "M10 Combo tool" for the M1 Garand. Works the same way in the M14 as the M1. To use it, it's helpful to have a cleaning rod section. It's kinda hard to describe without pictures, but here goes.
You'll want to keep the rifle level or a little muzzle down during this process to keep the firing pin inside (or, lower the hammer, is it a good idea to do that if the bolt is open?)
TO disassemble M14 bolt while in the rifle:
First, thread a cleaning rod into an M10 combo tool (for the M1 Garand)
1) Insert the tool into the chamber. One end has a screwdriver blade, the other has a flat ear protruding from one side. You want the screwdriver end to go into the chamber first, with the flat ear pointed down and to the right nearest the bolt and the cleaning rod sticking up out of the top of the action.
2) Slowly close the bolt on the tool so that the ear goes underneath the bottom of the extractor. You will probably need to help it the last 1/16" or so to compress the ejector spring and push the ear under the extractor.
3) Place your hand over the action, or place your thumb on top of the extractor and rotate the cleaning rod counter-clockwise (from the buttstock end of the rifle) to pop out the extractor.
4) Lift out the extractor and extractor detent/spring. Keep the muzzle at a somewhat downward angle to keep the firing pin from falling out of the rear of the bolt.
5) *Retract the bolt slowly, and you will see the ejector and spring on the left side. You can remove them now, or skip this step entirely if you're only replacing the extractor.
6) To remove the firing pin, or if it falls out, you will need to remove the trigger group. There is no need to seperate the stock from the action.
1) *Insert a .30-'06 case (empty, of course) into the chamber. This case has the same head dimensions as the 7.62NATO, but is longer.
2) Insert the firing pin (if removed) and ejector/spring into the bolt. Ensure the cut-out on the side of the ejector aligns with the hole for the extractor stem.
3) Close the bolt on the '06 case, pushing it home to compress the ejector spring so that the case head is flush against the boltface.
4) Install the extractor detent spring and detent.
5) Place the extractor stem into its hole. While pushing the bolt closed, press downwards on the top of the extractor until it snaps into place.You may need to use a small screwdriver or similar tool to compress the detent spring.
You can now cycle the action extract and eject the empty case. You should turn over the rifle and look in through the mag well to make sure that the bottom of the extractor stem isn't protruding from the bottom of the bolt.
*If you only change the extractor, without disturbing the rest of the parts, it may be easier to simply leave the bolt closed against the M10 tool. To put the extractor back in, you can simply rotate the tool back down (clockwise) and push in the extractor.