Chop into the trunk with an ax, making several gouges in the top.
Put on rubber gloves and pour herbicide with glyphosate into the ax gouges to begin to weaken the roots, eventually killing the stump.
Cover the trunk with a tarp and fasten into place by tying a rope around the trunk, cinching it tight. This will repel rainwater and keep sunlight from penetrating the stump, starving the roots.
Leave the trunk covered between four and six weeks, then remove the tarp.
Use the posthole diggers to unearth the topsoil near the trunk to expose the roots around the fence post. Pull the roots away from the fence post or cut them with a lopper or ax, as necessary.
Connect a tow strap to a truck and wrap the other end around the tree trunk. Pull the trunk out of the ground, away from the fence posts, removing the roots completely.