Outside of Apple customer support, an iPhone 5 screen can be fixed with a replacement screen kit. The screen can be installed alone for about $30 or as a joint assembly, with the speakers and home button, for about $60, as of 2015.
The first step in repairing a damaged screen is removing it by unscrewing the two star-shaped screws next to the phone's charger jack, pulling up on the screen with a suction cup, and using the appropriate flat-edged tool or a small razor to pry the screen free. While taking care to avoid bending the damaged screen past 90 degrees or damaging the ribbon cables, unscrew the three screws at the upper right corner, which hold the cable bracket in place, and disconnect the three cables still tied to the screen. If replacing the entire screen assembly, connect the new screen's wires, screw the bracket back into place, push the new screen down, and replace the final two screws by the charger jack. If the entire screen does not need to be replaced, remove the speakers and home button with the plastic tool before moving forward. Before taking on these tasks, note that repairs not done by certified Apple technicians may void the phone's warranty.