If we take their word for it, living in a body or with a soul is like Libya under Gaddafi: autocratic, terrifying, and full of torture. The soul definitely has more to say in terms of physical confinement—it's trapped inside a body, after all, which makes the dungeon metaphors more relevant. But the body also relies on political rhetoric with its accusations that the soul is a dangerous tyrant. (Back up to "In a Nutshell" to read about how Marvell's personal patriotism might be responsible for the ample dose of political imagery in this poem.)
Lines 1-2: The soul uses imagery of slavery and confinement to pluck the heart-strings of its audience. With the body described as a dungeon and the soul compared to a slave, who couldn't feel sorry for this poor little prisoner?
Lines 3-4: The soul adds to this gruesome picture more concrete images of confinement. Now we get an HD view of how it's enslaved. And, folks, this is no abstract slavery. This soul is locked up, with chains on both feet and hands. (Not that a soul has feet or hands—it's just a metaphor, silly!)
Lines 7-8: Why not add a bit of torture? The soul compares its life in the body to being strung up in chains.
Line 9: In case we didn't catch the torture imagery in lines 7-8, the soul makes it unmistakable in line 9. This life is nonstop agony.
Lines 11-12: Although the soul harps more on its physical confinement and agony, the body also goes in for some slavery imagery. Calling the soul a tyrant, it laments its status as a wretched, helpless servant, required to give in to the soul's demands.