All 4 Uses of

acquit

in

A Tale of Two Cities

I have a business charge to acquit myself of.†
Chpt 1.4

acquit = officially find "not guilty"

The friends of the acquitted prisoner had dispersed, under the impression—which he himself had originated—that he would not be released that night.
Chpt 2.4

acquitted = officially found "not guilty"

In the arrangements of the little household, Miss Pross took charge of the lower regions, and always acquitted herself marvellously.
Chpt 2.6 *

acquitted = handled (conducted or behaved)

No sooner was the acquittal pronounced, than tears were shed as freely as blood at another time, and such fraternal embraces were bestowed upon the prisoner by as many of both sexes as could rush at him, that after his long and unwholesome confinement he was in danger of fainting from exhaustion; none the less because he knew very well, that the very same people, carried by another current, would have rushed at him with the very same intensity, to rend him to pieces and strew him over the streets.†
Chpt 3.6 *

acquittal = official finding of "not guilty"

Definitions:

(1) (acquit as in: she was acquitted)
to officially find "not guilty" of criminal charges

Note that to be acquitted is not the same as being declared innocent of an offense due to the presumption of innocence in the American judicial system. The court determines if there is sufficient evidence to find someone guilty. Some crimes require a lot of evidence for a conviction, so while there may not be enough evidence to declare someone guilty beyond a shadow of a doubt, there also may not be enough evidence to declare a defendant innocent except through the presumption of innocence.
(2) (acquit as in: she acquitted herself well)
to handle oneself in a specified way -- which is typically in a positive way
(3) (meaning too rare to warrant focus)
Much less commonly and archaically, acquit can have other meanings. It can mean to release from a duty, as when Jane Austen wrote "I cannot acquit him of that duty" in her novel, Pride and Prejudice.

It can also mean to perform or complete an obligation, as when Charles Dickens wrote "I have a business charge to acquit myself of," in his novel, A Tale of Two Cities.