Study the great grunge bass players, like Krist Novoselic, Jeff Ament, and Ben Shepherd. Learn their songs and familiarize yourself with the scales, chord progressions, and picking styles of grunge metal.
Many bass players are taught to use their fingers when playing bass, but you should experiment with using a pick. Try several different types of picks until you find one that really works for you. Some players prefer the big bass picks, while others simply like to use guitar picks.
Leave the tone knob on your bass turned all the way up. Many indie rock, hip-hop, jazz, and pop bassists will turn their tone knob all the way down to give their bass a low, fat sound. However, turning the knob all the way up will give your bass a nice grungy sound that will more appropriately fit your preferred style of music.
Get a bass effects pedal, preferably some type of fuzz or overdrive. You won't want to use this effect on every song, but it will definitely help make some of your songs more unique and add an element of dirtiness to your tone.
Let loose when you perform live. A lot of what grunge was about was the recklessness and rawness of the Generation X'ers. Too many bands today sit on stage and try to play heavy music, but fake the attitude or don't seem to care at all. Remember why you love grunge metal and channel those emotions into your music and performances.