In the first stanza of this poem, the speaker seems restless, wandering through the streets of London near the banks of the river Thames. As he looks at the faces of those people he passes, he sees their sadness and their pain. This seems to suggest some link between the "charter'd street[s]," "charter'd Thames," and the depressed and miserable people of the city. This is further affirmed by the second stanza's reference to the "mind-forg'd manacles" worn by the people of London: the chimney sweeps, the soldiers, the prostitutes, and even the babies and the church-goers suggested by the third and fourth stanzas.
Thus, the word "charter'd" (in lines one and two) seems to suggest both confinement and having been mapped out (like how we might think of "charted territory"—land that has been mapped out). The streets and river seem to enclose or confine the people of the city, where their roles and lives have been "charter'd," or mapped out for them, so that they lack real freedom or possibility. The speaker says that "each [...] street" is "charter'd," suggesting the idea that, in the city, there are no places where there is room for imagination, nature, fantasy, or possibility. Everything is determined, regulated, and bounded, compounding the misery of the inhabitants and their sad lives, which inevitably continue in this way until they end up in the "hearse" of the final line.