In order to change your Player Name, tap on the Settings menu on the right-hand side of the screen, and then tap on "Player Profile" at the bottom. A screen will open showing your player name, mission progress, and power level. Tap on your player name(or the box that says "Enter Player Name", remove the "Enter Player Name" text, and type in the name you'd like to use.

Note that vulgar and inappropriate player names that go against the Community Guidelines could result in a ban from the game, so choose your player name responsibly!