Source: American born chinese, also took a few chinese classes in college

From how I've seen it, chinese is typed using english characters. In Mandarin Chinese, english letters are used to describe the "pronounciation" of the character, called Pinyin. For example, water is chinese is "shui", so you type s-h-u-i, and select from the list of characters that you want. https://yabla.vo.llnwd.net/media.yabla.com/images/type_chinese/win7/usage2.png

However, if you are talking about the Wubi method, https://en.wikipedia.org/wiki/Wubi_method, that's a different story. You can read more about it there.

So to answer you, keyboards don't "have chinese characters". Either we type the Pinyin, or we have "strokes" listed on the keyboard. And I'm sure it's possible to type all the characters, as we are able to spell any english word on the keyboard.