if you still have a residence permit as a student with the whole validity of more than 6 months (it is not important how long it will be valid after your asylum application, it must be valid when you make this application), you must not go to any asylum camp. You can (and even must after the German asylum law) make your asylum application in a written form. This rule is in §14.2 Asylgesetz, perhaps you can find a translation in Englisch. You just print the form from here:
fill it in and send it to the BAMF-Filial which is nearest to your place. After that you can normally stay in your place and make your studies further. You must naturally clear the question of financing if you have no own means. Normally you can apply for money for asylum seekers in the Socialamt in your place. Each asylum seeker can have social help from the state. But as far as financing is concerned I don't have so much knowledge.
You anyway need a good personal consulting. Look for councelling centers for refugees in your area. There are many in NRW.
If your residence permit is not valid anymore, you must make a personal application in one of the asylum centers. You can go to each one, but they will check the system and make a distribution. The most important criterium is what country you are from, because each federal state is responsible for a special number of world countries. And you can be sent to each federal state which deals with your country, depending on the quote Meike wrote you about.
So the written application is much better if possible.