The first four lines are an apostrophe. The speaker is addressing death, an inanimate thing, and personifying it, giving it human qualities. He tells Mr. Death NOT to be prideful because he thinks he is “mighty and dreadful”, a thing to be feared, because in the speaker’s view, he is neither mighty nor dreadful. Those that Mr. Death thinks he “overthrows” really are NOT dying, because Mr. Death has been overcome by God, who gives man eternal life. This is a Biblical view (the poem is referred to as Holy Sonnet 10), that death has been overcome by God. When we die, our lives do not end, but rather we wake up in a new place – death is really just like “one short sleep” and when we awake, we have a new life.
In line 8, we can infer that the speaker does not fear Mr. Death himself because all of the best men on earth, including him, will achieve not only “rest” for their bodies (bones), but “deliverance” of souls. Because Mr. Death has been overcome, HE will die, not man. When men do not fear Mr. Death, it kills him, in a sense.
Read about the poem here on eNotes.
One might paraphrase the first four lines of Donne's "Death Be Not Proud" in the following way:
Don't be proud Death, for although some in the past have called you
mighty and dreadful [something to be dreaded], you are not;
for those you think you overcome [kill]
don't really die, you mistaken fool, and you can't kill me, either.
The "mistaken fool" reference might be a little strong, but that's the general idea. Death is not powerful and is not to be feared, because those who die are not really dead.
The speaker apostrophizes death, speaking to it as if it were human, and reveals in the remainder of the poem what is behind his thinking and his bold statements: humans only sleep when they die, for they will "wake eternally" and death will exist no more--death shall be the one to die.
Donne's apostrophe leads to his paradox--humans don't really die when they die and, in the end, death is what will die.
Line eight simply contributes to the paradox: when men die they are only resting, and will later awake to their soul's salvation. The speaker's attitude, revealed in this line and the entire poem, is a bit sarcastic and flippant. He is disrespectful toward Death, and is putting it into its proper place, as he sees it.