Ulysses makes it clear early in the poem that he is bored with his sedentary life as king of Ithaca, and indeed with the responsibilities and drudgery of kingship. He longs to live out his days in the same way he has spent most of his life, in a spirit of adventure, seeking honor, fame, and new challenges alongside his faithful mariners:

My mariners, 
Souls that have tol'd and wrought, and thought with me- 
That ever with a frolic welcome took 
The thunder and the sunshine, and opposed 
Free hearts, free foreheads - you and I are old; 
Old age hath yet his honour and his toil; 
Death closes all:  but something ere the end, 
Some work of noble note, may yet be done, 
Not unbecoming men that strove with Gods.

Ulysses has been an adventurer all of his life, and now, in his twilight years, Tennyson portrays a man who does not seek leisure, but further challenges. By drinking "life to the lees," Tennyson means that Odysseus hopes to get as much out of life as he possibly can. Even though his body is old and fragile, Odysseus is still "strong in will," and believes that there are still adventures to be had.  For some people, a quiet, settled life might be something to aspire to. Not so for Odysseus.