It’s easy to hate Abigail Williams. Her flaws are blatant to the reader. She will do anything she can in order to avoid trouble. In act 1 of The Crucible, when she sees an opportunity to blame Tituba for their actions in the forest, she quickly lies, telling Reverend Hale that it was Tituba who called on the devil in the forest, asserting that she is innocent: “I’m a good girl!” However, when Tituba is eventually praised by Hale for turning her back on the devil and confessing her “crimes,” Abigail sees an opportunity. She too confesses her “crimes” and, like Tituba, begins naming names of supposed witches in the community, which starts the witchcraft hysteria. She knows naming names means danger for the named women, but she doesn’t care about their fates, only her own. She also knows that the women she has named are the lower-class women in society, and so they are easy for her to target.

From Abigail’s point of view, this shrewdness is a strength. It allows her to control the situation even though, as a young woman in Salem society, she would normally have very little control and power over anything. She also has great power over her peers. She bullies and threatens violence to make them all follow her.

Let either of you breathe a word . . . And I will come to you in the black of some terrible night and I will bring a pointy reckoning that will shudder you.

What are her motivations? How did she become so heartless and yet so powerful at such a young age?

We learn that Abigail is a survivor. At a young age, she witnessed the violent deaths of her parents. She tells her “friends,"

And you know you can do it; I saw Indians smash my dear parents’ heads on the pillow next to mine, and I have seen some reddish work done at night.

She is an orphan who has had to learn to survive in a dangerous world. She will do whatever she can to survive and, if possible, thrive, even if she has to lie and manipulate to do so.

And yet, one sees a different kind of strength in Abigail as well, something more than her coldhearted ability to control and calculate situations to her advantage. Part of the reason she is attracted to John Proctor is that he has taught her about hypocrisy.

I look for John Proctor that took me from my sleep and put knowledge in my heart! I never knew what pretense Salem was, I never knew the lying lessons I was taught by all these Christian women and their covenanted men! And now you bid me tear the light out of my eyes?

There is something to be pitied in these words. In one sense, we can see how a young woman could easily become infatuated by the powerful John Proctor. And there is a sincerity in her discovery as well, as she is shocked to discover that all of her conventional beliefs have been built on lies. She craves more of this knowledge. She can’t “unlearn” this, and this is part of her powerful desire for John Proctor, a desire that will lead to the downfall of her town (but not her downfall—Abigail Williams survives, as usual, escaping the town before it can catch her).

The original question had to be edited.  I would suggest that one of Abigail's strengths that becomes clear from Act I is that she is strong.  Abigail has a will that is able to wrangle the wills of others around her.  Abigail possesses a strong interpersonal intelligence throughout the first act.  She is able to understand how Parris is acting, and is able to suggest a course of action in which she and he benefit.  When she is with the girls, her strength as the leader is evident.  Abigail's strength in Act I is the strength that is within her ability to influence others.

Abigail possesses a two- pronged approach in her motivation.  The first is that she is looking to escape punishment for her actions in the woods.  She does not want to face the punishment from the town elders for what she has done.  When Mary Warren suggests that the girls tell the truth and accept the punishment, Abigail repudiates it.  A more subterranean motivation that Abigail possesses is that she covets John Proctor.  This becomes clear as the Act progresses.  The entire reason for her being in the woods was to cast a spell on Goody Proctor so that Abigail could become the next Mrs. Proctor. Abigail craves the affection she once perceived to have with John when they engaged in an affair.  In some respects, Proctor is her weakness in that the only time her machinations and scheming seems to take a break is when she is alone with Proctor.  Abigail's motivation is not small in any way.  She wants Proctor and will do what it takes to get him.

Outside of both of these elements in terms of her strengths and motivations, it becomes clear that Abigail is filled with weakness.  Seeing her parents violently killed, being devoid of any remourse or any authentic emotions, and carrying herself in a manner that reflects the world as a means to an end as opposed to an end in its own right are realities that fill Abigail with weakness.  The core of her personality, what Proctor would later call a "name," is something that Abigail lacks.  Abigail is more concerned with the temporal and the contingent as opposed to anything lasting and permanent.  This weakness becomes more pronounced as the drama escalates.  She will not stand for anything and thus ends up leaving Salem to reportedly become a prostitute.  Abigail's weakness in so far as a condition in which there is only human frailty evident is the projection she offers out of the First Act in the drama.