English Translation
No sooner had I come into contact with him than I determined to get to know him well.
Very unnatural. It turned out that the sentence pair is sourced originally from Tanaka Corpus (i.e. Japanese students translated English sentences), and I was very surprised. I thought this Japanese sentence was posted by a non-native speaker. There are two critical problems and two minor ones.
First, we don’t say 近づきになる (近づき as a noun + なる as a verb) . 近づき is used only in a form of
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As a noun: お近づきになる. You always need お (meaning 御). This is an expression. Don’t drop お from this context.
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As a compound adjective/adverb: 近づきやすい人 (easy to approach = friendly person).
お近づきになる as a common set phrase can be used only when the speaker looked up the other person or regarded him/her as a valuable person; and when the speaker is talking with the other person (i.e. having a conversation with the other respected/valuable person). For example,
あなたとお近づきになれて光栄です。（こうえいです） = I’m honored to be acquainted with you, Sir/Madam.
お近づきの印に、どうぞお受け取り下さい。（しるし、おうけとり）= I humbly offer you this token gift as a remembrance of our first contact.
Second, 知るようになろうと決心した is grammatically broken and sounds very unnatural. ～するようになる means “getting/becoming the condition of X”. It’s a progressive verb (i.e. illustrate the gap between two different time points). On the other hand, 決心した is in a past tense. It’s a one-time action. We don’t mix two different tenses like this into a single sentence.
私は彼をよく知ろうと決心した
私は彼をより深く知ろうと決心した
are correct and natural phrases.
Third, ～するや否や is a standard translation of “no sooner than”, but this phrase is too formal and less common.
Fourth, 決心する is a robotic word-for-word translation from “to determine”. 決心する
sounds much stronger in Japanese, and it doesn’t match in this whole context.
My alternative translation is:
彼と出会ってすぐに、私はその人となりを深く知ろうと思った。（かれとであってすぐに、わたしはそのひととなりをふかくしろうとおもった）
I replaced するや否や with すぐに, and 決心した with 思った.
人となり is optional but 人となりを知る is a frequent set phrase.
This is just a side note, but I recommend all of you to reach this article written by NHK (a quasi national TV broadcaster) – “What’s the difference between 近づく and 近寄る?”