Use only a few chords. The main chords used in a grunge song are the power chords. Those five chords can be used in combination to create a powerful sound that remains simple in construction.
Write lyrics that convey angst and dissatisfaction. Think about the dysfunctional relationships you've had in the past and mine them for song material.
Go heavy on the guitar. Grunge songs are mainly driven by the guitar. Occasionally play with the distortion pedal pressed. Choose a few sections of the song that will be best served by the distortion rather than using it throughout a song.
Add in a base line that is more complicated than the guitar parts. Write in a hard drum sound that is played throughout.
Sing the song with an emotional voice that occasionally makes the lyrics impossible to understand. As long as the audience can hear that something is being said, it doesn't matter if they can actually understand it. Use a low voice throughout. A voice that sounds like a growl is best.
Scream whenever it fits. A line of the lyrics can be screamed or the screams can be random. Many grunge songs start out with a lower volume and work their way up to screams at the end of the song.