Hello. Congratulations on scoring such a great score

Here is the criteria for counceling of neet2019

NEET Counselling 2019 will commence from the last week of June 2019 in online mode. State level counselling will be organized separately by the respective state authorities. Firstly, candidates have to register for counselling and fill their preferences for admission. 

Candidates clearing the cut off required by the respective colleges can apply for admission in that college through counseling. Check NEET Cut Off

Counselling for 15% All India Quota seats will be administrated by Medical Counselling Committee of the Directorate General of Health Services.

Seats will be allotted to students as per their merit, category, and availability of seats in colleges

Candidates who wish to resign from the allotted seat after the second round of NEET 2019 will have to participate in the online resignation process.

Once a seat will be allotted, candidates have to report to the allotted college within the notified schedule.

The seats that are left vacant after the first round of NEET 2019 counselling will be included in the second round.

Good luck!

First of all you need to put all the details of yours while applying for counselling, and I'd say while applying for counselling apply for both all India as well as Bihar . As you are saying you have scored 559 marks, you have major chances of getting good government colleges of your choice.  Just apply for counselling at both the portals and wait for your call. Good luck.