Looks like you've already accomplished it with arrays, what's the problem? Each class would be a row of the array and the same row in the player's array.
"Previously I had a character selection screen and an array listing all the stats for each class, and when a class was selected, those stats were injected into another array for the personal character. But saving and storing info of multiple characters is out of my league"
Surely the only options in the player array are either the data is filled out (they have made a character) or it's blank (they haven't made a new character and prompt this when they go to create one)