I think that no matter how you personally feel about dogs, we can all agree that dog costumes are hilarious. Yes, it's true that we are laughing at them, and not with them, but as long as they're willing to play along with it I don't think there's necessarily anything wrong with that.
It's actually pretty easy to make a costume for a dog, just as long as you keep a few basic concepts in mind.
Step 1: Know Your Limitations.
By this I don't actually mean YOUR limitations. I mean the dog's. Let's face it: your dog does not really want to wear a costume. Dogs don't really want to wear anything at all. To demonstrate, here's a video clip of my dog, the first time he ever had a collar put on him. He doesn't understand what it is and doesn't entirely understand how his own foot works, but he knows he doesn't want this thing on his neck. So please keep in mind what your dog can comfortably tolerate when you come up with your costume idea. If your dog will not wear a hat, you should not come up with a hat-dependent costume.
Step 2: Start With the Basics.
A simple, standard pattern for a dog costume has only three basic pieces. Hats are not necessary, nor is covering up any areas that you probably shouldn't be covering up, if you catch my drift. An additional huge bonus to this pattern is that you don't have to force your dog's arms or head into sleeves or neckholes; it all attaches together around him while he just stands there.
Use some tracing paper or scrap fabric to make your pattern pieces. The first piece is a "coat" that covers the back and torso; second, a "cumberbund" that attaches the sides of the coat around the belly, and the "neckpiece or shirtfront" that attaches the sides of the coat around the front (and also down to the cumberbund if desired for your design). Use a tape measure to determine the proper dimensions of these pieces for your dog.
You will permanently attach one side of both the cumberbund and the shirtfront to a side of the coat. Pick a closure device to attach the other sides - I've always used velcro, but buttons or snaps are fine if that's what you have. If you have a full shirt front you'll want to add another closure connecting the bottom to the center front of the cumberbund.
Step 3: Design Your Costume Around the Basic Structure.
Now that you have the structure, make up your costume to fit it! It's not as limiting as it might seem, even without hats.
Say you want to make a St. Patrick's Day costume. How about Oscar Wilde? OK, let's find a picture on Google. Now you know how to embellish your pattern pieces!
Henry the 8th? Same deal. Sure, the little hat might sell this outfit a little better, but add it to a group costume and the hat suddenly seems unimportant.
Historical figures aren't all you can make with this simple pattern. How about a New York Taxi? Driver's license optional. How about a subway car?
You can even dress him as a legendary dog - just add heads.
All of these costumes have the same basic structure.
Step 4: Bedazzle to Your Best Ability.
Design, embellish and bedazzle your basic pattern pieces in whatever way works best for you. Obviously it helps if you can sew, but it's not entirely necessary. It's not like the dog is going to need to wear this outfit for extended periods of time and for years to come. You can get the same results, more or less, using glue, staples or even duct tape. Just make sure whatever materials you use are not too heavy and do not chafe the dog or make him uncomfortable.