A stitching awl is a handle with a large needle in it, and a place under the needle to mount a small spool of thread. The thread is brought up through the eye of the awl, which is in the pointy tip of the needle, just like on a sewing machine needle.
Draw out a length of thread through the eye of the awl long enough that you could sew the whole seam with a running stitch. Thread the end of this drawn out thread into a tapestry needle. Put the tapestry needle through the seam (poke a hole with the awl if necessary) and draw all the loose thread through after it.
Take a stitch thus: push the point of the awl through the seam where you want the stitch to end. Draw back on the awl just enough to form a loop in the thread. Put the tapestry needle through this loop, and draw all the loose thread after it. Pull the awl back out of the seam (don't yank, just pull it out). Gently tug the thread on both sides of the seam until the intersection of the two threads disappears inside the seam. On both sides of the seam, only the thread traveling along that side should be visible.