I've done the dish soap mixed with shampoo before for buildup. Works really well, but make sure to use a damn good conditioner afterwards. You can seriously dry out your hair like that, and fixing that will be a PITA.
If I were you, just splurge for a bit of a higher-end clarifying shampoo. I know Back to Basics has a citrus clarifying shampoo that runs about $9 last time I checked- not all that much more than drugstore, but much better than using dish soap for your hair.