Depending on the felt thickness and how it fits on your sewing machine - it could be hand or machine stitched. If by machine use a long stitch and loosen the tension so the thread can travel the depth of the felt.
Hand sew with perle cotton or button-hole thread.
You could also use metal grommets to make a hole and slide a handle/s through that.
You could get a leather handle with tabs and sew the tabs onto the top of the bag. - use a leather needle -either hand or machine.
You can also sew a non-felt lining in a sturdy fabric - attach handles to the lining and stitch the top edge of the bag to the finished top of the lining. That way there is very little stress on the felt.
The last is what is done on this purse - it is a fulled sweater recon, but the same concept for the handles. The zipper and handles are attached to the lining only.
HTH - you have lots of options. Choose one that works the best with the size, wt and intended use. Best wishes.