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.
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 --- 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...?