This depends on how obvious you want to make it. I'm going to give two suggestions, of which the latter is by far my favorite, but is also less subtle.
Option 1: Just describe what a speaker says in the three different levels.
Don't draw special attention to it, but write dialogue differently than you normally would by giving more statements of the speaker without making clear whether they actually said them out loud.
"I wanted to serve my kingdom", he said. But his words also said "I actually wanted to escape from the boring life at home" and, more hidden and covered up, "I am scared to death of becoming a fucking loser like my father".
It seems likely that something strange is going on here, but it's hard to pinpoint. It will certainly make this character interesting because they seem to know so much about everyone they talk to. Personally, I think the problem here is that it will never be clear whether this is simply good intuition and knowledge of human beings or whether it is something more.
Option 2: Introduce new words to help the reader and to enhance your story.
The special character you are talking about likely needs to categorize these different types of hearing they possess in some way. You can use this categorization as a guide for the reader that also clues them in that something special is happening in the scene.
"I wanted to serve my kingdom", his loudvoice said. But his selfvoice said "I actually wanted to escape from the boring life at home". They both tried to drown out his truthvoice, which said "I am scared to death of becoming a fucking loser like my father".
Here the text uses categorizations as they are used by the special character (who is definitely the POV in this case). I have introduced "loudvoice" as the description of the normal speech of the speaker, "selfvoice" as the personal perception of the speaker, and "truthvoice" as the actual, potentially repressed voice of truth.
Something like this can really help to flesh out the character. For example, you mention that in some cases people simply speak the truth. Well, what would the character then hear? They would hear three voices at once, saying the same thing. This might be beautiful and rare for them.
Example (in which the special character is named Warren):
The man in handcuffs raised his voice. "I don't deny it", he said. "I killed them all, I butchered them in their sleep. I loved it, and I would do it again." His loudvoice, his selfvoice and his truthvoice all spoke in unison and formed a perfect harmony. It was beautiful. Warren suppressed a blissful smile.
Personally I believe this is an interesting way of doing it, because it allows you to write about a new dimension - namely how this all affects the main character, how having this ability might have changed his priorities concerning what people say. You don't even have to mention dialogue this way at all, you can sometimes simply focus on how the different levels make your character feel:
His loudvoice was a bore, but his selfvoice was so fascinating! Warren could not stop listening.
Option 3: Don't listen to me, do your own thing.
This might be my most important advice: this part of your novel seems to be important, and it might be one of the stand-out features of your novel. So think about an interesting way to portray it yourself. It's not difficult to get the point across; it's difficult to make it exciting and unique. And it will be better if it is something that is truly yours, not just something you copied from advice on the internet.