How do I tell how many times I've died?
In the original Spelunky, there was this neat little room that told you exactly how many times you have attempted to beat the game, along with how many times you've died doing so.
In the HD Remake, there is no such room.
How can I tell how many times I've died in the HD remake?
In the main menu, press "Player Stats" and you can see your attempts (plays), deaths and wins as well as your last run, average gold and most gold earned.