I wander thro' each charter'd street, Near where the charter'd Thames does flow, (1-2)
We get a double dose of confinement here. The streets and the Thames? The poem announces its obsession with the theme early on, and implies that somebody (he doesn't say who) is trying to tame or confine nature itself—here represented by the Thames.
And mark in every face I meet Marks of weakness, marks of woe. (3-4)
While we don't have any specific descriptions of confinement here, the language is super-restricted. The word "mark" occurs three times. The repetition of "mark" is also similar to the repetition of "charter'd" in the first two lines.
In every voice, in every ban, The mind-forg'd manacles I hear. (7-8)
Manacles=confinement. There's no question about that. None. What. So. Ever. The phrase "mind-forg'd" makes it seem like confinement is totally a mental thing that's got nothing to do with literal chains and charters. Our own minds are able to enslave us. Spooky.
How the Chimney-sweeper's cry Every blackning Church appalls; (9-10)
The chimney sweeper makes us think of confinement in several ways. First, we have the image of some little kid climbing down a chimney—that's a pretty confined space if we ever saw one. Second, lots of these chimney sweepers back in the day didn't have a whole lot of choice about whether or not they wanted to work.
And the hapless Soldier's sigh Runs in blood down Palace walls. (11-12)
All soldiers are "confined" to certain rules and regulations—they've gotta do what they're told, more so than other people. The repetition of the S sound all over the place ("hapless," "Soldier's," "sight," "walls," "runs") reflects the restrictions of a soldier's life.