Whoso list to hunt, I know where is an hind, But as for me, alas, I may no more. (1-2)
The "hind" is a richly symbolic animal that has a long history in Greek and Roman mythology. It is also an important part of Petrarch's Sonnet 190, which Wyatt imitates. Wyatt's speaker indicates that Wyatt is actually chasing his literary ancestors, but feels he cannot chase them anymore. Perhaps he feels that he cannot "capture" the greatness they have. Or rather, maybe he thinks he needs to find a different subject.
The vain travail hath worried me so sore, I am of them that farthest cometh behind. (3-4)
The speaker is talking about coming last in a group of hunters, but he could just as easily be talking about being last in a group of poets—coming later in history. The poem is obsessed with the tradition of the sonnet, and with Petrarch in particular.
Yet may I, by no means, my wearied mind Draw from the deer, but as she fleeth afore Fainting I follow. (5-7)
The deer is an important literary figure, as we've seen (associated with mythology and with Petrarch). The speaker is obsessed with it, which is like saying he's obsessed with literary history. And it's killing him. He wants to be free from all this dead weight, but just can't keep his "mind" away.
And graven with diamonds in letters plain There is written, her fair neck round about, (11-12)
The deer is associated with death. The word "graven" means written or inscribed, but we can't help hearing the word "grave" in there as well. The speaker should be careful about chasing this deer, which is another way of saying that, if he cares about his writing career, he should be careful about regurgitating old literary symbols.
'"Noli me tangere, for Caesar's I am, And wild for to hold, though I seem tame"' (13-14)
The deer belongs to Caesar and should not be touched. Perhaps this whole bit about not touching means "find something else to write about, Thomas Wyatt, the deer belongs to Greek myth and Petrarch."