I'm trying to say "I think I can speak a little Japanese because I have studied a lot".
My translation attempt:
Does it sound right?
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We don't do translation checks, so I'm just giving a number of pointers
you're using AからB to try to construct "A because of B", but in fact it translates to "B because of A", so you need to switch the statements to get the intended causality. This is the same for other conjunctions with a similar meaning, like ので (see When to use ～ので vs ～から).
なった思う is no good. なった can be used to end a sentence or to modify a noun. To say that you thought something, you need to use the quoting particle と to give なったと思う.
勉強【べんきょう】することがある is grammatically correct, but makes it sound like you are counting how many times you have studied Japanese. Here it might be better to use (the correct conjugation of (see below)) 勉強している "I'm [currently] studying" or "I have been studying".
Since your main clause is in the past tense (ようになった), the cause needs to be at least as old as the effect, that is, use the past tense here: 勉強していた. Alternatively, you could say
or, (trying to improve on your old construction)