I think you are mixing things together.
The MySQL.user table, are for the DBMS to handle users of the database, not of a custom application, so unless you are trying to create a DBMS manager, you shouldn't access any of MySQL's own tables.
If you wanna store you passwords in the database encrypted, you simply encrypt them in your code, and then they will also be encrypted in the DBMS logs and in the DB.
But as the docs state, you show NEVER allow access to the DB logs, to other then an administrator and the DBMS itself.
Thanks for the clarification.
That means that there is no need to connect to the DB using my user's credentials, correct?
I can use my admin UN/PWD to connect the DB, and just verify my users are legit against my users table?
Yes, but it's not good practice to use your administrators account to connect to the database, but as long as you are developing you can do it, but should you ever go into production, then you would typically create a MySQL user for each app, and only give that user to the app's tables.
But for now don't think about this.
Besides this your solution sounds okay.