When I press the steam button, brown juice is getting out of my iron (I suppose salts with water?), soiling the clothes that I try to Iron. The bottom part of the iron is clean but it seems that there is something inside that needs cleaning. What part of the iron do I have to clean and how?
The brown coming out with the steam is mainly due to rust in the steam passages. Given the price of a new iron, the sensible way to deal with it is to replace the iron and use only distilled or reverse osmosis filtered water in the new one, to prevent recurrence.
If that's not a viable option for some reason, it might be possible to soak the iron's steam system with a rust remover. I think it's unlikely you'll find anything that will take rust out of the steam passages that won't itself stain clothes. Mechanical cleaning would require something similar to a pipe cleaner, but thin enough to push through the steam passages (which are far from straight, to hold the water inside the sole plate as it heats and boils).
Another option (one we used when I was a child, forty-plus years ago) is to keep the iron dry, but spray a light mist of water directly on the clothing as you iron. You can use tap water for this in most locales, because the solids in the water won't get concentrated by repeated boiling in the iron's sole plate (but test first on a piece that's already got stains; if you have high iron content in your water this can also cause problems). This last also works with a dry iron (one without steam passages), which is how I recommend using it if you decide to replace the iron.
You can prevent (slow down) deposits by using distilled or reverse osmosis water.
You don't have a clean button but I would try using the steam button as a clean button. Fill it up, heat to max, unplug, take it to the sink, and hold the steam button util it empties. It may not empty but worth a try.