Decide what you wish to write your song about. Many people prefer to start their songwriting with a general topic. Think about this, staying within the generally somber and angst-ridden realm of most grunge music. Possible sources of inspiration are people's tendency to selfishness or arrogance, and the moral and intellectual bankruptcy of society as a whole.
Brainstorm ideas if you can't think of a topic. Anything can inspire a lyric, such as a line from a book, a magazine article or even another song. Open your ears to all of the possible sources of inspiration around you. Take note of any line you come across that makes you think. Alternatively, you can attempt to capture a particular emotion through lyrics.
Open your song with an interesting line. The first line in your song is vital to hooking your listener. Think about the first line from "Am I Inside" by Alice in Chains: "Loneliness it shadows me, quicker than darkness." Use lyrics like this to draw the listener into the song, and to establish atmosphere. Introduce any character or story as clearly as possible with the first line if your song takes the form of a story.
Follow though with lyrics that develop your story or main idea while maintaining the song's tone. The general tone or mood of the song can be carried through words of a particular affect. For example, Soundgarden's "Black Hole Sun" uses words such as "black," "disgrace," "stench" and "dead" to clearly establish a somber and bitter mood. This theme is generally fitting of the grunge genre.
Devise a "hook" for your song. "Hook" is a musical term, more commonly associated with pop music, which essentially means the catchy chorus line. Grunge music does include hooks, but these are usually more groaning and angst-ridden than pop hooks. Come up with a simple, one- or two-line hook that will stick in people's minds. For example, the chorus to "Am I Inside" is "Black is all I feel, so this is how it feels to be free." The line is both memorable and evocative, as it cues listeners to ponder its exact meaning.