So far, Poussey's crime is unclear. What we do know about her past is that she lived in Germany, where her father was stationed with the US Army. She had a romantic relationship with the base commander's daughter, and this is what got her family reassigned to the States.
In a season four flashback, Poussey is back in America visiting her friends. She gets separated from them and has a wild night on the town, befriending some drag queens and hitching a ride from bicycling monks who turn out to be members of an improv group.
By the end of season four, we find out that Poussey is in prison for being busted over a small amount of weed.