Use gain or effects. The Soundgarden style wasn't called "grunge" for nothing. The rhythm guitar is a crunchy, muddy track, churning out often slow-moving power chords. Try to get this kind of tone on your axe using effects pedals or the settings on your amplifier.
Set up chord tempos. The Soundgarden guitar usually follows the drum track, sounding along with the beat. Listen to songs like "Black Hole Sun" to hear the guitar drop along with each percussion bar in a steady, easy-to-follow rhythm.
Stick to a "major-key" guitar melody line. Soundgarden guitarist Kim Thayil was adept at making non-minor note sequences sound melancholy, or "gloomy" according to Soundgarden critics. The dark quality of the music is based more on its listing tempos and slow groove than its notes.
Use repetition. Going over and over a riff is trademark Soundgarden. You can hear it in prominent hits like "Fell on Black Days" (Superunknown) or "Pretty Noose" (Down on the Upside)--the guitar repeats itself as the background noise increases, for a distinctly unnerving effect.
Check out Soundgarden tabs. Tablature of Thayil's solos shows you the nuts and bolts of a Soundgarden lead track. From there you can build a comprehensive idea of what a Soundgarden riff sounds like.