How can you see how many lives you have in Minecraft?
In Vanilla Minecraft, you have an infinite amount of lives, so naturally - There is no limit on how many deaths you can do*.
In Hardcore Mode however, you only have one life, and that life is determined whether you're still playing or staring at a Delete World button.
For mods, such as Hardcore Questing Mode, you can find your remaining lives in the front page of the quest book.
*Assuming you're not using any command blocks or mods.
As for the number of times you have died, you can display it in the scoreboard like so:
/scoreboard objectives add Deaths stat.deaths Deaths /scoreboard objectives setDisplay sidebar Deaths
or you can find it in the Statistics menu.