SED, KPSS, or any other communist party in Eastern Block weren't open. If you wanted to join, best way would be to get active already in teen years, in party's youth organisation (Komsomol in USSR). Then your way to party would be quite easy. Otherwise, if you were a valuable student, worker or just citizen, you could be approached by party official in your factory, university etc., and invited to join. Of course you still could join on your own initiative, but it would be a little harder. In any of these cases, your background and your family background would be checked; you'd have to write an extensive CV; and have two party members to support your application. And even then you'd become a full member only after at least two months of "candidacy" period, and still could be denied.

Of course, these rules varied depending on country and period, but what's above should give you a general idea.

There's a little bit of discussion in I am a young person in Communist Europe, how do I make a career in politics?, but in it /u/Visceralrealism's comments overlap somewhat with /u/pothkan's comments here.