Post-colonial criticism focusses on texts which deal with the domination and oppression of one group by another, usually on racial/ethnic grounds. This most often takes the form of writings about empire, but can also be readily applied to literary texts such as American slave narratives, as these also involve the domination of one group by another.
Post-colonial criticism would focus on the theme of oppression which of course is extremely prominent in this work by Douglass. It would discuss the ways in which the slave-owners maintain their often brutal power over the slaves in this story and how society as a whole is complicit in this. It would discuss the portrayal of oppressor and oppressed (for instance, in this particular text, some of the slave-owners are depicted as being nothing less than pure evil). Post-colonial criticism would also look at ways in which Douglass, as a member of the oppressed race, manages to resist and eventually escape slavery, to tell his own story; in post-colonial terms, this would be cited as an example of a marginalised group finding its own voice.