verbs - I have been beginning

No problem when "I began to learn English in 2000". But what if "I began to learn English from 2000 till now more times without success".

Verb begin something that happens in a moment, not something continuously. After that does this correct:

I have been beginning to learn English since 2000, but never continuous consistently, tenacious till now.

Answer

If you're intentionally meaning to say that you've been starting to learn English many times over the past few years but never really getting much further than starting, you can certainly use the form you have! I've used it myself this way (though not when talking about learning English).

I've been beginning this project for five years and I'm just now finally getting into it.

Often, we will emphasize "beginning" when speaking to show that our past efforts haven't gotten much beyond the beginning stages.

So, in your sentence, you could say:

I have been beginning to learn English since 2000...

I'm sorry to say, though, that the rest of your sentence could use a bit of work. I don't know that we generally use "tenacious" in the way you have it. It's a lovely word but I don't think it works here.