Several of Jesus' parables give the message that it is not those we expect who will "go to heaven" (with all the caveats on the suitability of that phrase!).
The story starts off with a rich man, he dressed well and lived well. Now this was not the problem, being rich is not a problem. It is what we do with riches that can be a problem, the attitudes it engenders in our hearts. Each day a beggar named Lazarus was laid at his gate. So everyday the rich man had an opportunity to help a poor man, but did not do so. It is perhaps also significant that the poor man has a name. In fact I think he is the only person in a parable who gets a name. In doing this Jesus is showing that the poor man was actually the more "important" person in God's eyes.
In time both men died. No wealth, status nor power can prevent us from dying. Lazarus was taken up to Abraham's side, while the rich man was taken to Hades and was suffering torment. The rich man still had the proud attitudes he had acquired on earth and called out to Abraham to send Lazarus to ease his pain. What we do on the earth and the characters we develop are crucial, they will have eternal significance.
Abraham tells the rich man why he is suffering, and also that no one can pass from Hades to Heaven or vice versa.
Then the rich man asks Abraham to send Lazarus to warn his family to change their ways. People always complain that God does not make His message clear enough. This is garbage. The Jews had all the Law and the Prophets to tell them the truth but ignored them. Jesus walked among them and did many signs and miracles, yet few believed in Him. It is a lie to think that if only someone sees miracles they will believe. We are responsible for our unbelief. Even Jesus rising from the dead did not convince many of them. The human heart can be very hard.