Remove the toilet by unscrewing the bolts on the base of the toilet with a wrench and lifting the toilet off the flange. The flange is the flat-headed pipe installed under the toilet.
Remove the flange from the concrete by prying it up and off the concrete, or by chipping away the concrete around the flange with a jackhammer. Chip away the concrete over the drain pipe connected to the flange with the jackhammer for about a foot. If necessary, cut away the elbow joint of the flange out of the drain pipe with a pipe saw to remove the flange.
Cut a channel in the concrete from the old location of the toilet to the new location, with the jack hammer. Make it wide enough to lay the 1 1/2-inch PVC drain pipe.
Connect the PVC drain pipe to the existing pipe and run it up to the new location for the toilet. Connect the flange elbow to the drain pipe. Take the drain apart once it is plumbed, and reassemble it using PVC cement on the ends of the drain parts. Let the drain dry for at least eight hours.
Fill in around the drain pipe with a bed of gravel. Mix mortar and trowel it over the drain pipe to fill in the channel. Let the mortar dry for 24 hours before walking over it.