I'd like to give it my best until the very last moment.
I'd like to perfect this right up until the deadline.
I want to make this as perfect as possible right up until the deadline.
ご質問ありがとうございます。
①
"I'd like to give it my best"＝「私の最高を尽くしたいです」
"until the very last moment."=「最後の瞬間まで。」
②
"I'd like to perfect this"＝「私はこれを完璧にしたいです」
"right up until the deadline."＝「締め切り直前まで。」
③
"I want to make this as perfect as possible"＝「これを出来る限り完璧にしたい」
（②と同じ）
ご参考に！
アメリカ合衆国 
