A sounding hammer/axe or increment borer are two non-destructive methods used to determine "soundness". Both of which, I doubt, you have in your possession.
However, you can use the non-cutting portion of an axe to "sound" the tree. Firmly, but not destructively, tap the outside of the tree to determine whether or not you hear a hollow-like sound. This simple method is employed by most foresters and timber buyers when looking at standing timber for sale.
If you are worried about the structural integrity of the tree in question, then I suggest contacting a certified arborist.