It's all OK apart from the last verb form. It shouldn't be a conditional - the 'would' clause is at the beginning of your sentence, so the clause following 'when' should be in the past. I'd rewrite it as: I realized I would be able to claim that I really know this language when I could understand him.

I think the thing that threw me with the sentence is the second "would". Tatiana - I'll presume you're a Russian speaker, and thinking of the Russian глагол НЕсовершенного вида (imperfective verb) - понимать (to understand) rather than понять (to understand), and you're using it (rather than the pf) because you want to communicate regularity in the past? If so, just stick with "understood" rather than "would understand"; the regularity is communicated by the first "would". I hope this helps!

Nothing necessarily wrong with tenses there, but the sentence sounds more logical if you say "could understand him" instead of "would understand him", since the focus is on your ability to understand, rather than other factors.

Yes, it's right.