This question tests modifiers.
What later became the famous novel 'Uncle Tom's Cabin'? The magazine series.
Therefore, the portion of the sentence after the first comma must refer to the 'series', and hence, begin with 'which'. Thus, options A and D are out.

This is the sequence of events:
HBS wrote the magazine series -> It became a famous novel
When we want to show tense sequence for 2 past events, we use the past perfect to refer to the earlier event, and simple past to refer to the later one. So, the use of 'had later become' in option E to refer to the second event is incorrect.

In option B, the issue is with the last portion. HBS's mission is two-fold: portray the impact of slavery and further the abolitionist cause. This is not brought out clearly in option B.

In 1851, what HBS wrote was a magazine series. From this perspective, its transformation into a famous novel happens in the future. To view a future event from the past, the use of 'would' is correct, and C is the best answer of the lot.

If your question is 'is the use of would necessary?', I would say no. The sentence can also be written as:
In her 1851 magazine series, which later became the famous novel Uncle Tom's Cabin, Harriet Beecher Stowe sought to portray the impact of slavery and further the abolitionist cause.

However our job is not to rewrite the sentence in the best possible way, We just need to pick the best answer choice of the lot.