Rafi was right on a 3 tier application.
But if you don't have a middle tier in your application then ..
Yes. Using the init method to lookup the DB conn is a good way as that is the method first called in the life cycle of a servlet and during the life time it will not be changed.
Another good strategy would be ti initializa a pool of DB connections rather than a single one. This will give better performance.
Again initializing the pool can be done in the init() method()