Fitting an inner tube to a bicycle wheel requires it to be placed between the rubber tire and the hard metal rim. This sounds easy but there are a few steps that need to be executed in order. By performing the procedure in this order, you ensure the new inner tube is positioned correctly and the air stem helps secure the tube to the tire and rim.

Unscrew the valve cap off the inner tube. If it is a Presta-style tube with a thin, narrow and long stem, unscrew the rim nut along with the cap.

Coat the inner tube with some baby powder. Use the powder to give the inner tube viscosity when placing it into the tire. Inflate the inner tube about one-quarter full. Give it just enough air from the pump to provide the shape of the inflated tube.
 
Push the inner tube into the bicycle tire. Push one edge of the tire onto the carbon bike rim and push the tire edge up to the inner edge of the rim. Spin the tube until the air stem is next to the rim's stem hole. Push the bicylce stem through the hole and screw the rim nut onto the stem if using Presta tubes. If using Schrader tubes, just push the stem through the rim hole.
 

Push the other tire edge onto the rim so the inner tube is wedged between the tire and rim. Use the curved end of the tire lever to push the hard rubber tire edge into the rim to secure it into position.

Inflate the inner tube to its recommended pressure with the pump. Screw on the stem cap once the tube is full.