I have caught Ho-Oh, beaten the elite four and Red, and caught Lugia but I still cannot find out how to get past the purple doors in the Ruins of Alph. I would greatly appreciate it if someone would tell me how to. Thank you.
I'm pretty certain you can't open those doors. The buildings are just there to get in the way ;)
You can however, go through some of the buildings that have open doors in the front or side. (There are some other secret places too, which I will put in a separate question.)