For the timing part you can use the millis() function. Here is a tutorial that may help.
An Arduino output pin cannot drive an injector so an external circuit is necessary (like your relay). A MOSFET is a good choice. Here is an injector driver circuit. Choose a MOSFET that has a logic level input and can handle the current required by the injector. The diode is necessary to protect the MOSFET from the high reverse voltage when the injector is turned off.
