If I wanted to say, "Ken's getting better and better at Japanese", then would this be correct?
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Your usage of だんだん is just fine. It's a standalone adverb which can be used without any other particle. An adverb is usually placed just before what it modifies (in this case, じょうずになる).
You have to use the progressive form of なります; otherwise the sentence would mean "Ken gets (or will get) better and better at Japanese".
To make this a bit more natural, you can add a subsidiary verb ～てくる and say: