A number of misconceptions with your question:

1. Horrendous as the GULAG system was, it wasn't a system of extermination camps like some of the Nazi camps - many millions of people (including people imprisoned for political reasons - and ordinary criminals were sent to the GULAG as well) served out their sentences and were released afterwards.

2. After they completed their sentence, they were subject to internal exile for life in remote regions like Central Asia or Siberia (but in 1957, such exile was revoked, after de-Stalinisation). An ex-prisoner's family would be forced to join him. Let's take the Nobel prize winning novelist Alexander Solzhenitsyn as an example. Thus, in 1952, after Solzhenitsyn completed his 7 year sentence for writing disrespectfully of Stalin in a letter to a friend, his wife divorced him rather than move to Kazakhstan. They remarried in 1957 after he was allowed to return to European Russia.

3. Finding a job was not an issue - the government assigned you to a job, maybe an unpleasant, dangerous, dirty job, but you had a job. Solzhenitsyn (who was a graduate of Rostov State University) worked as a miner, bricklayer, and foundry foreman.

4. Or were the records just not there to do a background check like today? No, this isn't how a totalitarian government works. Every single person had to have an internal passport (you needed permission even to visit different parts of the country) and all of the documents you had accumulated throughout your life would be on your permanent record which would follow you around for ever. OTOH, after deStalinisation it was officially acknowledged that Stalin had unjustly repressed many people and Solzhenitsyn became a schoolteacher.