How to know when chicken breast has cooked through?
A thermometer is the only way to be sure.

For methods with a consistent level of heat (stove, oven), you should able to learn the average cooking time, and outwards cues of color and texture that match the right internal temperature. Remember that these will vary with cooking method and temperature. Internally, the meat should look opaque and white.

These are CUES of doneness, not guarantees. Use the thermometer.

For methods with inconsistent levels of heat (bbq, campfire), wait until it looks nearly done and then check with the thermometer.

Err on the safe side. Prepared properly, chicken is a lot more forgiving of overcooking than steak.