Find the hidden signal in one of the Rat Man's dens
This can be gained in Chapter 2 - Test Chamber 6.
When you make your way across to the button, press the button and GLaDOS will dump some garbage instead of a cube. The Garbage will come towards you (as long as you keep your two portals on the angled platforms). Stand in front of the garbage so it hits you. Shoot a portal to where you started and grab the Radio. Now look at where the Angle platforms are where you need to shoot the portals to get to the button you. If you look up will see that you can shoot your portal on the wall to the left of the angled platforms. Shoot a portal as far as you can to the right. Now shoot another portal on the Angled Platform CLOSEST to you. Use the Aerial Faith plates and you will be launched into a secret room. Go to the desk and place the radio on the desk. Stay in that room for a bit and the achievement will unlock.
You can see where the Secret room is if you stand beside the button and look at the angled platforms. Look to the right and up a little and you will see a broken window which leads to the secret room.
Do this achievement before solving the puzzle so your last autosave is still at the beginning of the chamber. I had to restart the chapter several times because I didn't consider this.
When launching the garbage, use the white panels opposite the chamber entrance as the exit portal to send it back towards the chamber entrance and avoid the radio landing in the acid.
Be careful and make sure to angle down while the faith plate shoots you through the window or you can lose it on the frame, then it will fall in the acid. Also, it may get lost behind the pipes when you land. Duck down and keep trying to grab it from underneath. It will eventually dislodge.
@ 15 Nah, it's supposed to be the last remaining survivor from the Aperture Science employees who has gone insane and speaks to a Companion Cube. Search up Valve's comic Lab Rat to get some more info on him.
#7 is a pretty good explanation. It took me a few tries, but I got it last night. I ended up just launching the garbage onto the inclined panel area without involving any portals. The radio happened to be one of the items that did not bounce into the acid.
Today, I had another idea: before launching the garbage, put one portal on the first inclined panel and the second on the wall panel opposite the Rattman den. That way, the garbage should all shoot directly into the room and you won't have to worry about the acid. Of course, the window into the room won't have the funnelling effect of a portal, so it may not work. I'll try it myself tonight.
A good suggestion to avoid dropping/losing the radio while being hurled into the den is to face away from whereever you're going, this way there's no risk of the radio getting stuck anywhere or you dropping it against something.