You need to file your I-589 with the USCIS Asylum Offices in your area. Once the application is file you will be called for an interview. The process usually takes around 1-2 years.
Asylum is filed with the BCIS within one year of entering the US. The BCIS makes a decision to grant or deny. If you are illegal you'll be referred to an immigration judge. You'll go to court and the immigration judge will then decide on your asylum application. Should he deny it, you can appeal to the Board of Appeals.
It is a long process if the BCIS denies it the first time and very very expensive and exhausting. You must be very certain of your facts and have all your paperwork in line.