Can I define my own characters in a programming language such as C# or C++?

That's not going to work in the long run, because for longer messages the algorithm will run out of characters.

In the meantime, just provide the program with an array of characters and manually list all the special characters you want to use. Then make the program increase the index by one whenever it encounters a new word.

Again, if you just want to practice programming, go ahead. If this is supposed to be productive, go back to a per character encryption.

Sure, it's called a dictionary or hash. In Ruby...

a = [1, 2, 3, 4, 5]

b = ['a', 'b', 'c', 'd']

Hash[a.zip(b)]

=> {1=>"a", 2=>"b", 3=>"c", 4=>"d", 5=>nil}

You can replace any of those symbols (left or right) with any other symbols depending on the language. With the C family of languages I'm not so sure. But, if a person were smart regardless of language, they would triple encrypt it, meaning encrypt the message and the dictionary too along with the code to encrypt and decrypt.

Most likely so if you include the 110,000 unicode characters covering 100 scripts. And if you sub two unicode characters instead of one for each word you increase the "vocabulary" size by a very large factor. You are not defining new characters. Here is a sample I hope they show: Δ, Й, ק, م, ๗, あ, 叶, 葉, and 말.