The best engineering college at Chennai offering program in marine engineering is the Indian Institute of Technology, Adyar, Chennai which offers a 5 years full Time dual degree B.Tech + M.Tech program in Naval Architecture and Ocean Engineering. Admission in the course is based on JEE Advanced scores.
Regarding other institutes, that depends on the acdemic profile of the author. Some of the DGS approved maritime training institutes at Chennai are as follows -
1. AMET University
2. School of Maritime Studies (Vels University)
3. Indian Maritime College
4. Hindustan Institute of Maritime Training Post Sea Training Centre
5. Indian Maritime College
There are many more. A list of all such approved colleges is made available on the website of the Directorate General of Shipping - http://www.dgshipping.gov.in/.