Does my female character need to "date" Alistar in order to make him king?

If he dies in the final battle, you aren't queen yet, so you can't reign. You aren't married to him. If you both survive, the text at the end says that you marry him six months later. Anora stays queen if he dies.

The Dragon Age wiki says that you can make a political marraige even if you aren't in a romantic relationship with Alistair.

I don't see where it says anything about his personal quest, except that certain outcomes require him to be hardened. Staying with him as a lover, not a wife, requires hardening, (pun unintended), and a few other options won't happen unless he is hardened.