The first stanza reads more like a soliloquy than a dramatic monologue. (A soliloquy is a speech where a character is alone on stage, or believes himself to be alone, and speaks his thoughts out loud to himself.) Ulysses begins:

It little profits that an idle king,

By this still hearth, among these barren crags,

Match'd with an aged wife, I mete and dole

Unequal laws unto a savage race,

That hoard, and sleep, and feed, and know not me (Lines 1-5).

These lines are not kind to the island of Ithaca, Ulysses’ wife, or his people—Ithaca is described as “barren crags,” Penelope is called “aged” (which implies she’s barren as well), and Ulysses’ subjects are described as “savage” and bestial. Because Ulysses would not say these cruel things out loud to anyone else, he is likely alone and talking to himself.

While not as cruel, the rest of the first stanza also reads like a reflection, not something Ulysses would say in front of an audience. Moreover, while the opening five lines meditate on external things—the island, Penelope, and the people of Ithaca—the remaining 27 lines of the first stanza focus on Ulysses.