If you are applying for asylum in the United States, and have no other basis for staying here legally, then you must wait until your asylum case is approved before applying for a green card (U.S. lawful permanent residence). For most people applying for asylum in recent years, the application process moves fairly quickly (unless you are denied asylum in immigration court and are appealing your case). If and when you are approved for asylum, you'll be able to apply for a green card one year later.
If, however, the processing of your case has been delayed or is stuck in the appeals process, then it makes sense for you to look for an alternate way to obtain a U.S. green card. For example, you may marry a U.S. citizen, win a place in the diversity visa lottery, or find a U.S. employer willing to sponsor you. There is no prohibition on having more than one type of application pending with U.S. Citizenship and Immigration Services (USCIS) at the same time.
While asylum will offer you a long-term right to stay in the United States, it is not as secure as having a green card. Your right to asylum is conditioned on your continued fear of returning to your home country. Once you have a green card, however, this condition no longer applies, and you can start counting the time until you are eligible to apply for U.S. citizenship, the most secure status of all.
In fact, if you are approved for asylum and apply for a green card on that basis, you may (particularly if your home country has gone through a dramatic improvement in its political situation) need to prove your continued fear of persecution at that time. No notification is sent to an asylee to remind him or her to apply for a green card, so you should keep track of the one-year anniversary of your asylum award, and apply for the green card (adjustment of status) as soon as you're eligible.
If it looks like you won't be approved for asylum anytime soon, however, then it makes good sense to pursue some alternate path to a U.S. green card.
Immigration law is highly complex, and submitting a winning asylum case requires detailed presentation of facts and legal arguments. If your case seems to be tied up in processing, perhaps it's because you need some help in making a convincing case for asylum. Consult an experienced immigration attorney for help with both the asylum application and any other applications for a green card.