If you are applying for a U.S. green card through VAWA (the Violence Against Women Act) -- that is, applying for lawful permanent residence without your unhelpful, abusive U.S. spouse or parent -- then you will indeed have to provide documentation of that abuse.
If you have not already been placed in removal proceedings, then the filing process begins with filing an application by mail (Form I-360 if you are just starting the process, or Form I-751 if you are a conditional resident seeking permanent residence). After that, you may be required attend an interview at an office of U.S. Citizenship and Immigration Services (USCIS). If your application is denied, you might be placed into immigration court proceedings, at which time you could renew your application for a U.S. green card.
For more information on the eligibility rules, see Domestic Violence Against Immigrants: VAWA Protection.
First, it's important to understand the types of abuse that can qualify someone to self-petition for a green card under VAWA. The definition is broader than you might think. It includes not only physical violence, but threats, intimidation, economic abuse, social isolation, sexual abuse, verbal abuse, harassment, and more.
With that in mind, the self-petitioner in a VAWA case will need to gather credible, detailed evidence that he or she (or his or her child) has experience such abuse by the U.S. citizen or lawful permanent resident (LPR) spouse or parent. Here are some possibilities for putting together this showing of abuse:
This list is not exhaustive. You may be able to think of additional ways of proving abuse. Also, see The Self-Petition Under VAWA for information to help people draft affidavits on your behalf.
Realize, however, that proving a VAWA case also involves gathering documents proving other matters, such as the citizenship or LPR status of the petitioner, the existence of a good faith marriage, and more.