The personification of Death in this poem is the conceit around which the poem revolves. Capitalization has long been used in poetry in tandem with personification. By capitalizing Death, it becomes more than simply an abstract idea; it has become a proper noun, like a person's name. Capitalizing Death, as one may capitalize God, gives the word more prominence and supports the imagining of Death as a sentient being with motivations and understanding.
Ultimately, however, although Donne characterizes Death as engaged in an ongoing attempt to "overthrow" humanity, the point of this poem is that the speaker thinks Death is not "mighty and dreadful" at all. While Death can certainly take away our lives unexpectedly, it is actually a "slave to fate, chance, kings, and desperate men." Donne is saying that we ourselves can summon death, by way of poison or war. All death really does is put us to sleep, after which "we wake eternally," so, therefore, how is death any stronger than "poppy or charms" (meaning drugs, or spells to induce sleep)? Just as we wake from an ordinary sleep, we wake from the sleep Death puts us into, and so Death himself is vanquished by the hope of eternal life.
A note on the concluding couplet: it is almost certainly by chance that these two lines no longer rhyme in modern English. Remember that Donne was writing in the early seventeenth-century, when the pronunciation of English was different. "Eternally" would originally have rhymed with "die." In certain parts of northern England and Scotland, this is still true—the word "die" is pronounced "dee." In Donne's case, it's likely that he pronounced "die" in the modern way and pronounced "eternally" as "eternal-lie," based on the part of the country he was born in and its regional accent.