Looks like you've got a couple of problems here. First, I doubt you have the mysql server installed. The command you typed should have a listing like mysql-3.23.56-1.9, or some similar version. The first thing I would do would be to download the server rpms and go from there. Which leads me to my second point.
It looks like the RPM you've got there isn't for the MySQL server, but for php-mysql (I think that is a client adapter which allows php/mysql to work together). The dependancy failure you've listed tells me the mysql(client) isn't installed, or it's not installed in the place the rpm is expecting it. My guess is the former, and that you probably need the mysql client libraries installed (I think this is mostly driver/interface material). You'll need to do this after the server install from the first paragraph.
After these steps, you should have all the pieces in place. Good Luck.