TAKOBOTO

became
I became able to speak japanese

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だから、えいごがうまくはなせるようになりたいです。

So, I want to become able to speak better english

どのようにしても、わたしはかまわない。

I don't care whatever you do.

２１せいきがどのようになるのかそうぞうできますか。

Can you imagine what the 21st century will be like?

にんげんはあかんぼうのときはよつんばいで、それから２ほんのあしであるくようになり、さいごはとしをとってつえがひつようになる。

It crawls on all fours as a baby, then learns to walk on two legs, and finally needs a cane in old age.

あなたがはやくよくなるようにいのっています。

We are hoping for your quick recovery.

ように has a meaning of "to do" or "so that one can do" which I guess would include "become." The first example can literally be translated as "For that reason, I want to do so that I can speak better English."

My second example follows that vein.

bamboo4

Um, this might be of use but it's especially for the youni NARU -sentence pattern. But it said in the comments of the "youninaru" that this grammar entry would do, so...One of the possible usages of ように:

ようになる expresses a change that takes place gradually.

With verbs:

Vdic + you ni naru= "reach the point where~"/"come to~"

子供は自分で部屋を片付けるようになりました。
(My) child has reached the point where she cleans her room herself.

Vneg + you ni naru= "reach the point where ~not"

林さんはたばこを吸わないようになる。
Mr.Hayashi has reached the point where he does not smoke.

*Vdic= Verb, dictionary form

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Besides なる, there's also する and 言う verbs that can be connected with ように to create more specific meanings...Did some of these have their own entries...?

Naino