In order to get admission for Merchant Navy you need to appear the entrance exam like IMU CET conducted by Indian Maritime University, JEE Advance, CUSAT CAT etc. You can check the eligibility of the exams with given below links.

IMU CET Eligibility

CUSAT CAT Eligibility

JEE Advance Eligibility

Top institute for Merchant Navy you can prefer:

Marine Engineering and Research Institute, MERI Kolkata
Marine Engineering and Research Institute, MERI Mumbai
Lal Bahadur Shastri College of Advanced Maritime Studies and Research, Mumbai
T.S. Chanakya, Mumbai
National Maritime Academy, Chennai
Indian Institute of Port Management, Kolkata

Good Luck!