These underlined parts are both grammatically correct, but their meanings are subtly different.

"I would really like to..." This is probably the one you want to use. This means that you want to do something A LOT!

"I really would like to..." This is a little different. In English, we use the word "really" as a question sometimes. If you say, " I would like to..." Someone might say, "Really?" This is the same as saying, "That doesn't make sense! Is that true?" You can then respond, "Yes, I REALLY want to..." This is the same as saying, "Yes, It's true! I actually want to!"

Also, this is the correct way to say your sentence: " I would really like to improve my speech in English." Or, "I would really like to improve my ability to speak in English."

"Doubt" is not the correct word in this context. The word you need is, "question."

I hope this helps! Good luck!

I really would like to improve my speak in english?

I am looking forward to improving my spoken English very much.