The speaker of "Elegy Written in a Country Churchyard" is a thoughtful, pensive guy. He likes to be alone. At night. In graveyards. So that he can think about death. Good times. But you know the type, right? You might find someone like this speaker in your local coffee shop, wearing all black and maybe just a tad too much eye makeup, reading Camus or Sartre and thinking deep, deep thoughts.
But there's more to this speaker than his arguably morbid tendency to hang out in graveyards. He wants to make sure that we all remember the lives of people who lived before us, even the lives of simple, country folks like the ones buried in the churchyard where the poem takes place. He wants to be conscious of the way that he himself will be remembered after he's dead and gone, and that means thinking carefully about how other people see him now.
Sure, this might seem morbid, but the speaker seems to want to set himself apart from the kind of rich, snobby people who just care about erecting huge monuments and mausoleums in their own honor after they die. Instead, he wants to leave something less concrete behind him in the memories of the people that he cares about.