How do I clean a very dirty refrigerator?
There's a lot of good advice here on cleaning the inside of the fridge.  One thing I'd like to add:

Don't forget to periodically clean the fridge's coils. The coils are typically found underneath or on the back of the fridge, and are meant to dissipate the heat.  The coils will, over time, get blocked with dust etc. and the fridge runs less efficiently as a result of such buildup.    Keep the coils free of dust and you'll save money and extend the life of the fridge.  Use a vacuum with appropriate connectors, and be careful when moving the fridge, vacuuming the coils, etc.

Here's a good reference: How to Save Energy--Clean Your Fridge Coils