How to make a complex novelty cake at home?
If you want a sculpted cake of this type: 



then I would advise against trying it at home. This is a skill you have to learn over years. Basically, you have to be an experienced sculptor in working with cake materials, which are harder to use than the typical art materials like clay. Look up "cake sculpting" on YouTube to see what they are doing. Or search for Cake boss episodes, taking into account that this "reality" TV is heavily edited - they are showing you a very clean and quickened version of the process. Really, I think that 500 dollars is cheap for having your drawing sculpted. By the way, the airbrushing tool alone used for finishing the cake in the picture runs at around 200 dollars, although of course you don't strictly need it for making the cake in such a shape. 

You seem to be good at drawing. If you can make a finished, colored drawing out of it instead of just a sketch, you can go to a bakery who has a foodgrade printer, and have it printed on the cake. They mainly advertise it as "your cake with your photo on it", but of course they can print a digitized drawing too.



If you don't want a picture, it is also doable as a 2d picture piped onto a flat cake. If the 500 dollars are for a sculpted cake, then having the picture piped can be much cheaper (I don't think that they would have required 500 dollars just for piping, but who knows, maybe you live in Manhattan). It would be this type of cake, but with your drawing on it: 



If you have steady hands and some artistic experience, this kind of decoration is a skill that can be learned in maybe 20 hours. Take a piping bag, make icing, and practice on other surfaces first, even stretched plastic wrap. When you are good enough, attempt the drawing on a practice surface. Then you are ready to make the cake. Fondant or marzipan will look best as the background for the piping, but they are very hard to work with. If you have never done it before, use a cake frosted with buttercream as the background. If you have the time and inclination, you can learn how to make marzipan flowers and add them as 3-d elements on top of your drawing. 

The basic skills you need are: 


bake consistently good bisquit layers. You can cut off domed parts, but it is a hassle, and if you are not good at getting the leavening right, you are risking cutting into a metallic tasting cake after all this work. 
Making buttercream. It is not hard, but requires a bit of exercise and a few tools. You should be accustomed to baking by weight, know the feel of butter and eggs at the right temperature and the right stage of beating, etc. If you have never done it before, again, make 2-3 batches of buttercream before you try the actual birthday cake. 
Frosting cakes. Your drawing will probably require a large rectangular cake, which is harder to get evenly frosted than a small round one on a turntable. 
I already covered the piping. Marzipan flowers are a further skill I mentioned, but optional. 


The Internet is full of tutorials on decorating cakes, watch YouTube videos about the more visual parts. There are good articles on making icing too, and we also have one or two older questions on that.