First of all, you should consult an attorney rather than seeking the comments of strangers on the web.
As a general matter illegal aliens can be granted asylum and eventually green cards if they meet all the normal requirements for asylum. Of course, if you are denied, you will have brought yourself to the attention of the DHS and they will place you in deportation proceedings.
Also, you must apply for asylum within one year of your arrival in the United States. If you have been here for more than one year, then you cannot be granted asylum unless one of the very limited number of exceptions apply. For example, changes in country conditions can excuse a late filing.
If you have real persecution fears and cannot receive asylum because of a late application, then you can still receive something called withholding of removal, which is far less desirable than asylum but does give you temporary legal status here.
Again you are advised to obtain legal counsel.
Good luck to you.
