How can my aunt get custody if she was deported?

This issue is complicated because it deals with the intersection of family law and immigration law. You (and/or your aunt) need to speak to an attorney who has significant experience with both areas of law immediately.

Your question seems to imply that the child is still in the US, but under whose care & control? Since they are apparently not married and there have never been any judicial proceedings, there isn't much your aunt can do from Mexico against the father. But, it appears from you question that the boyfriend may or may not be the father and may or may not have the child, so it is not clear how or why he might try to "take her daughter away from her now".

The fact that your aunt has been deported adds a layer of complexity to the case that will probably requiring getting an immigration lawyer involved. Let's assume that your aunt was still in this country. I also have several questions about the case. You describe the father as your aunt's boyfriend. Were they ever married? When the child was born was a declaration of paternity filed by the father? If they were never married and if there was no declaration of paternity, then, the father has no rights to the child until there is a determination of paternity by an appropriate court. Otherwise, your aunt would need to begin a parenting plan action. All of this is going to be more difficult because your aunt has been deported. That is why you will want to consult an immigration lawyer in addition to a family law attorney.

Under Washington law, even though your aunt was deported, if Washington was the child's home state within the last six months, the courts here still have jurisdiction to determine a parenting plan for the child, including which parent the child should reside with. Your aunt may need to file a legal action here to have a parenting plan established for the child.

I am confused. Where is the child? Who currently has the child? I am going to assume the child is in Mexico with mom. It would be very difficult to force the Mexican government to disgorge the child.

There is a lot of missing information here. Why was she deported (did she commit a crime which led to the deportation)? Is there an order for custody and visitation now? Is her boyfriend the father and has paternity been established? If you are in my area and are looking for an attorney for your aunt, please contact me for a free consultation.