NASA's Tumblr breaks down the process:
For astronauts, the voting process starts a year before launch, when astronauts are able to select which elections (local/state/federal) that they want to participate in while in space. Then, six months before the election, astronauts are provided with a standard form: the "Voter Registration and Absentee Ballot Request–Federal Post Card Application."
'Space voting' was first used the same year it was implemented in 1997. NASA astronaut David Wolf became the first American to vote in space while on the Russian Mir Space Station. Current space station resident astronaut Shane Kimbrough is the most recent astronaut to take advantage of the opportunity.
So remember: If astronauts can make time to vote, so can you!