Four. That's how many times this phrase occurs in the first stanza. It is a simple yet forceful way of saying "I am alive. I exist. I'm still here." Nobody cares about the speaker, and nobody knows what he's about. It's almost like he's dead, except—he's not. This is why he keeps saying "I Am." He is letting everybody know that he's still alive and well.
Line 1: The speaker says "I Am" twice in this line. The first is just a simple statement about existence, while the second implies that the speaker has a specific identity that interests nobody.
Line 3: The speaker tells us more about what he "is": he is the sole consumer of his woes. He doesn't actually eat his woes, so consuming is a metaphor for his relationship to suffering.
Line 6: Here again the speaker uses the phrase to make a statement about existence. He also uses it as part of the simile in which he compares himself to "vapours tost."