For what I assume is the intended sense, OP's example #1 is the most natural alternative:
1) I teach you [how] to speak English
But the second version doesn't have that meaning:
2) I teach you speaking English.
In principle that could make sense if we interpret it as meaning I speak in English while teaching you [something, not necessarily how you should speak English]. Or at a stretch we could interpret "speaking English" as a noun phrase equivalent to "spoken English", but that's a rather non-standard usage.
Example #3 is simply ungrammatical. There's not much more to say about it.
Example #4 is syntactically "valid", but if the teacher was any good at his job that's probably not how he would phrase it. He'd probably use the format of example #1 anyway, but if he wanted to focus on the actual subject (the spoken language), as opposed to the goal (that you learn to speak English), he's more likely say:
5) I teach you spoken English
(Note that this version more strongly implies I only teach spoken English, not the written form.)