Translation:He thinks he is somebody, but he is nobody.
That is not good English.
Here, in English, "thinks" is being used as a transitive verb: you're conveying what he is thinking. He is thinking what? He is thinking that he is someone. The subordinate clause "that he is someone" serves as the object of the verb "thinks."
As a matter of usage, we can (and often do) drop the conjunction "that" to yield "He thinks he is someone."
In "He thinks to be someone, but he is nobody," you're missing information. Who or what is to be someone? He? She? The dog? His neighbor? From the Dutch sentence, we know it is "hij." So we need to put "he" into it. Yes, we already have "He" at the beginning, but all that tells us is that "He" is the one doing the thinking. We need it again, and then we need to conjugate "to be" to match "he." "He thinks that he is someone, but he is nobody."
This can get confusing, I know, when you consider other verbs. "He plans to be someone," "He hopes to be someone," are fine, but those sentences speak of some future act that he is planning or hoping to do (or, in this case, to be). This is different from stating a present fact that he thinks (that he is someone). If you wanted to talk about a fact that he hopes is the case right now, you would have to do the same thing you do with "thinks": "He hopes that he is somebody."
I hope. I hope what? I hope that I have not made a mess of this explanation.
In English, "that" can be omitted when the subordinate clause follows the most common reporting or thinking verbs.
He thinks [that] he is somebody.
She said [that] she was coming to the party.
They believe [that] it's going to rain tomorrow.
I heard [that] this is a very good movie.