SEGA Mega Drive Ultimate Collection

Use your magic as soon as you have any, don't save it up as the achievement will take longer to get. You should be able to use your magic 10 times by the time you get to the 3rd level. Look out for the little guys in blue, especially in between levels, hit them to get the blue bottles of magic.

tip- go to settings and set lives to the highest

Quick way- use every magic bottle you get straight away while traveling though, doing this you can get the achievement around mid level 2

Slower way- just go through as you would normally and attack every little blue dwarf as quick and as many times possible, you should get 2-3 out of each one, now when you get up to harder enemies like giants and armored ones use magic, going through like this you can get it during level 3

OR if you just want it as quick as possible without going through levels, Your magic attacks carry over between lives and continues. As you have 3 lives per credit, and you start with 4 credits, you can simply use the magic you start with, and then allow yourself to die. repeat until you have used magic 10 times.