This is less of a CAP related question. However, it's one I actually can answer, so I'll point you in the right direction. More or less,
Saboros, you have to have some level of programming skills and access to your own server/client in order to go about it. For documentation on how to set up your server, go here:
https://github.com/Zarel/Pokemon-Showdown. For documentation on how to set up your own client (you'll probably need access/space on a reliable host as well, which probably will cost money), go here:
https://github.com/Zarel/Pokemon-Showdown-Client. And if you want to know how to actually code up Pokemon and formats/tiers, etc in Showdown's language, go here:
https://gist.github.com/dtalley11/8789538.
Now, since that's all out of the way, I have to stress that this is not the place to ask further questions on the subject. Talking to the showdown coders when they aren't busy is probably your best bet to actually understand something if the links I've given you don't help. Short answer to your question however is yes, you can have your own Pokemon Showdown server/client and play around with your own fakemon there (example of one in existence is Fuji Labs, as wh0sy0urpapa mentioned). However, CAP is not the place to talk about your own personal creations.