We've had some great discussion on this already, but let's see if we can break down HOW to answer this question in the quickest and easiest way we can! Before we dive in, here is the original question with the main differences between each option highlighted in orange:

Although Alice Walker published a number of essays, poetry collections, and stories during the 1970s, her third novel, The Color Purple, which was published in 1982, brought her the widest acclaim in that it won both the National Book Award as well as the Pulitzer Prize.

(A) which was published in 1982, brought her the widest acclaim in that it won both the National Book Award as well as the Pulitzer Prize
(B) published in 1982, bringing her the widest acclaim by winning both the National Book Award and the Pulitzer Prize
(C) published in 1982, brought her the widest acclaim, winning both the National Book Award and the Pulitzer Prize
(D) was published in 1982 and which, winning both the National Book Award and the Pulitzer Prize, brought her the widest acclaim
(E) was published in 1982, winning both the National Book Award as well as the Pulitzer Prize, and bringing her the widest acclaim

After taking a quick glance over the options, a few key differences stand out:

1. which was published / was published / published
2. and / as well as
3. brought / bringing

The best place to start is anything that will knock out either 2 or 3 answers right away, so let's work our way backwards on our list and start with #3: brought vs. bringing. This is clearly an issue with verbs, so let's determine which one is the best to use here.

To begin, we need to find the subject and verb. Here is the original sentence:

Although Alice Walker published a number of essays, poetry collections, and stories during the 1970s, her third novel, The Color Purple, which was published in 1982, brought her the widest acclaim in that it won both the National Book Award as well as the Pulitzer Prize.

If we cross out all of the modifiers (and there are a few), we're left with this:

Although Alice Walker published a number of essays, poetry collections, and stories during the 1970s, her third novel, The Color Purple, which was published in 1982, brought her the widest acclaim in that it won both the National Book Award as well as the Pulitzer Prize.

Once you take out the modifiers, which are mainly there to confuse readers, it's clear that the subject of the sentence is "The Color Purple." And what did The Color Purple do? It brought Alice Walker acclaim! Let's look over each option more carefully to figure out which ones use the right verb to match the subject. I'll still cross out any modifiers to help you see things more clearly.

(A) The Color Purple, which was published in 1982, brought her the widest acclaim in that it won both the National Book Award as well as the Pulitzer Prize --> OK
(B) The Color Purple, published in 1982, bringing her the widest acclaim by winning both the National Book Award and the Pulitzer Prize --> WRONG
("The Color Purple bringing" isn't the right verb.)
(C) The Color Purple, published in 1982, brought her the widest acclaim, winning both the National Book Award and the Pulitzer Prize --> OK
(D) The Color Purple, was published in 1982 and which, winning both the National Book Award and the Pulitzer Prize, brought her the widest acclaim --> WRONG
(While this does use the correct verb "brought," it also tries to add another verb "was published" inside a modifier, which is a big no-no because it creates a comma splice! So, let's rule this one out too.)
(E) The Color Purple, was published in 1982, winning both the National Book Award as well as the Pulitzer Prize, and bringing her the widest acclaim --> WRONG
(This one is wrong because it uses the incorrect "bringing" and it has a comma splice before "was published.")

We can eliminate options B, D, & E because they use the wrong verbs or have comma splices. Now that we're left with only 2 options to choose from, let's tackle #2 on our list: as well as vs. and.

Whenever we see the word "both" joining two objects, those two objects MUST be joined by the word and:

Both X and Y = CORRECT
Both X as well as Y = WRONG

(A) which was published in 1982, brought her the widest acclaim in that it won both the National Book Award as well as the Pulitzer Prize
(C) published in 1982, brought her the widest acclaim, winning both the National Book Award and the Pulitzer Prize

There you go - option C is the correct choice because it uses the correct verb "brought" and uses the idiom "both X and Y" correctly!