I was trying to use mysqli and I noticed that it didn't work. I really don't think it's smart to use already deprecated functions but would mysql work?
The MySQL functions are just deprecated to point out it's cooler to use MySQLi, you should still be safe to use them.
You need to install the MySQL extension manually. The pocketmine installer only installs the sqlite extension. I woud advice you to use the pdo functions to make and handle your database. With pdo you can easily switch between SQLite, MySQL, MySQLi, Oracle, ...