Is this different from 5e? Afaik in DnD you stand up after 3 successful deathsaves, could be wrong though.
In 5e, you only stand up after you have, at least 1 HP.
That means that a downed character (0 HP) will stand up if receives any healing effect, or rolls a 20 on a death save.
Otherwise, after you have 3 sucessful saves you are stabilized and will regain 1 HP after 1d4 hours (thus standing up).