The Grim Reaper isn't selectable from the outset. I completed the story mode with another character, then from the Main Menu selected Story Mode>New Story and Grim was available.
As for completing story mode as The Grim Reaper, it should be relatively easy as you get to keep your uprgrades from your original play through, the character doesn't matter as you just seem to be upgrading one universal kart as it were. Therefore this playthough will help toward the slight grind that is:
Just one thing, dont overlook the upgrade for handling/aerodynamacy, some of the corners in this game are sharp and I was struggling with a particular course near the end untill I upgraded this skill, then I came first next go.