Check the metal wheel for any sharp areas which could damage the new tire. File them down with a metal file.
Insert the new tire's valve stem through the hole in the metal wheel. Screw a valve stem tool onto the valve stem. Grasp the valve stem with a pliers and pull it to make it tight in the hole. Unscrew the valve stem tool.
Lay the new tire flat on the ground. Angle the metal wheel into the tire and push it in as far as possible. Insert a large flathead screwdriver between the tire and the wheel. Use the screwdriver to pry the tire over the wheel's rim.
Flip the tire and wheel over. Mix 1 part dish soap with 3 parts water in a spray bottle. Spray the mixture onto the new tire to make the rubber more pliable.
Ask a friend to help hold the tire on the ground. Use one hand to push the tire's inner edge nearest to you down as far as possible. Insert the large flathead screwdriver between the tire and metal wheel. Use the screwdriver to pull the tire underneath the wheel's rim. Move 2 to 3 inches down and pry that portion of the tire underneath the rim. Continue until the tire is underneath the rim all the way around the wheel.
Bounce the tire and wheel a few times on the ground to settle the two together. Wrap a ratchet strap around the middle of the tire's tread. Tighten the strap until the tire beads seed on the wheel's rim. Remove the ratchet strap.
Tighten the valve core into the valve stem. Use an air compressor to fill the tire to the appropriate psi. Screw the cover over the valve stem.