I am looking for a cheap ($50 or less, although if needed my max price range could be $70) rgb backlit keyboard that moves colors in a wave fast like the Razer Blackwidow Chroma. Mechanical is preferred but I'll take what I can get. If you don't know what the Razer Blackwidow Chroma wave mode looks like, go here: http://www.razerzone.com/gaming-keyboards-keypads/razer-blackwidow-chroma#clicky as you can see it moves extremely fast across the keyboard. Thanks! :)
First thing first, this keyboards are overall really well built, solid and reliable for a reasonnable price. However you will have a hard time founding what you are looking for in the 50-70 USD price range so the following keyboards are going to be in the 80-150 USD price range.
Now you know that, you want to choose if you need the numerous keypad on the right, you will chose Quickfire XT if you need it and Quickfire TK (both from coolermaster) if you don't. (It really come down to preferences but I think that for gaming the TK version is better and cheaper).
Now if you want backlighting you'll need to drop some more money for a TKi or a XTi, they are modable so you can download custom backlighting templates on the internet to customize the way the leds light (I don't have any link right now but I've seen keyboards moded just like you described it).
Unfortunately if the thing you're looking for really is a razer RGB like lighting you'll need to go to razer where the price is insanely high and the quality is much decieving but this is the gaming market right now.
So if you can't spend more than 100+ USD you will be forced to chose between fancy RGB and quality build :/ That's why I found coolermaster to be a good compromise between both.
EDIT: Fixed whole answer
As said before, to that price, you will not found something that satisfies all your requirements. A few month ago, I was in the same situation: looking for a RGB, mechanical, well built keyboard at a price as low as possible.
Obviously I had to make some concessions and I finally went to the Corsair Strafe. This keyboard is very well built (plastic, but good plastic) and is "only" 100€.
It's higher than your price range (like the Quickfire XT / TK) but at this price you have RED backlight that is fully customisable. It also comes with a numpad.
One more thing, becarful to the type of Cherry MX you choose (Corsair Strafe is sold in several versions in which the type of switch is different).
You'll notice that the Strafe also exists in an RGB version, at 170€.