This construction is used for non-exhaustive (incomplete) lists and has very much a casual feel to it and really isn't the sort of construction you'd use for what you're trying to say there.
"By next year" is 来年までに (and means before next year comes). I suspect you mean to say "by the end of next year", so something like 来年の内に or 来年の年末までに.
来年に means quite literally "next year" and will be understood to mean the period between January 1st and December 31st of next year....leaving out the rest of this year, you see.
Your verbs in English are things you hope to become fluent in...and that part doesn't make it into your translation (and it needs to be there). The able to part similarly is not present.
話す --> 話せる --> 話せるようになる
読む --> 読める --> 読めるようになる
For 分かる lets switch to either 理解 or 聴き取る
理解する --> 理解出来る --> 理解出来るようになる
聴き取る --> 聴き取れる --> 聴き取れるようになる
来年一杯までに日本語を上手に話せるように、読めるように、そして聴き取れるようになります。 would be my suggestion, although it would be much more 日本人らしい to substitute 頑張ります for なります at the end. (or なりたい)