I wander thro' each charter'd street, Near where the charter'd Thames does flow, (1-2)
Chartering something—controlling it through legal means, or making it more narrow—is a way of killing something. This becomes clearer in the poem as everybody the speaker meets on these "streets" seems dead.
How the Chimney-sweeper's cry Every blackning Church appalls; (9-10)
We can't think about chimney-sweepers without thinking of the color black, which, at least in a poem like this, definitely makes us think of death. Just to make sure we "get" it, the word "blackning" appears in the very next line. Death, death, death.
And the hapless Soldier's sigh Runs in blood down Palace walls. (11-12)
Soldiers fight wars, which makes us think of death. The image of blood running down a wall does as well. Taken together, the speaker clearly has war-related death and war-related violence on his mind. Strangely, these things are "present" in London, a place far away from any real battlefield. Wow.