"I will do everything what you want" is a literal translation of the Czech, but is not a good English translation. Some native English speakers would say it, but it's not considered standard English, it's more of a dialect way of saying it heard in some parts of Britain and the US. An English speaker would typically say, "I will do everything you want" or "I will do everything that you want."
"I will do everything that you want" is also correct. "that" is optional, but not incorrect if inserted.