I cannot rest from travel; I will drink Life to the lees (6-7)
Ulysses cannot stop moving; he is synonymous with perseverance and living life to the fullest. He will keep drinking that bottle until it's all gone; and we believe him because he says it with such conviction. He doesn't say "I want to" or "I should," but "I will," as if he were reminding us of his own "will" to persevere.
But something ere the end, Some work of noble note, may yet be done, Not unbecoming men that strove with Gods (51-3)
Even though death is approaching, Ulysses won't give up; he wants to do one more thing, but he's not sure what. It's a "work of noble note," which suggests he doesn't just want to go exploring, he wants to make a splash. He remains committed to doing big things in the world. He did once fight against the gods, after all.
Push off, and sitting well in order smite The sounding furrows; for my purpose holds To sail beyond the sunset, and the baths Of all the western stars, until I die (58-61)
Ulysses proves he means business; he orders his mariners to set sail, and he tells them to get those oars moving too. He has a "purpose" or goal that he wants to pursue – to sail really far away – and he's willing to leave behind a happy life in pursuit of that goal. Perhaps that is the "work of noble note" he refers to in line 52.
Made weak by time and fate; but strong in will To strive, to seek, to find, and not to yield (69-70)
Ulysses doesn't really know what he's striving for or what he's looking for; none of the verbs in the last line of the poem have objects, which makes us wonder what exactly Ulysses thinks he's doing. While a few lines earlier he had a "purpose," he now seems caught up in the moment, unclear about what exactly he wants to do. And we get caught up in the moment with him! The lines are so catchy that it's easy to overlook the vagueness of Ulysses' plans.