If you are an asylee in the United States, you can apply for adjustment of status to permanent resident (otherwise known as getting a green card) one year after the grant of asylum. That assumes you have also been physically present in the United States for one year's time.
You can also wait longer than that to apply, if you wish, or if you need to in order to meet the physical presence requirement. But the longer you wait, the higher the risk that conditions in your home country will normalize; in which case U.S. Citizenship and Immigration Services (USCIS) could decide that you no longer need the protection of the United States. In that case, it could not only deny the green card but place you into removal proceedings for possible deportation.
The primary application for the purpose of applying for a green card while in the U.S. (called "adjusting status") is USCIS Form I-485, Application to Register Permanent Residence or Adjust Status. However, this is just the first of a packet of forms and documents you'll need to submit, which include:
The completed Form I-485 must be filed with the USCIS office having jurisdiction over your place of residence in the United States. You can pay any and all fees with a single check or with a credit card (in which case you must also file Form G-1450).
USCIS will inform you the time and location of your appointment for biometrics (fingerprinting and so forth).
If your application is incomplete, USCIS could return the whole thing to you for refiling; or might follow up with a Request for Evidence (RFE).
Next, USCIS will call you for a personal interview. There, a decision will likely be made on your application (though USCIS might ask for further documents or follow-up instead). If you are approved, a green card will be sent to you some weeks later, and you'll be on your way to qualifying for U.S. citizenship.
If you want to apply for adjustment of status as an asylee, you might wish to consult with an experienced U.S. immigration attorney for help. That's especially true if your country has been in the news lately, with word of a civil war or conflict having ended or conditions having otherwise improved. The attorney can assist you with the entire process and accompany you to your adjustment interview.