All the usual disclaimers apply . . in short, do this at your own risk, blah blah blah. Also, people have different methods of doing this job, and I have mine.
I just replaced the backlight bulb on my climate control, and thought I'd share . . I've gotten into the habit of having a camera around all the time. This applies to the older E36 analog units. The replacement bulb is a 1.2watt Sylvania #2721, or an equivalent of another brand. You'll also need a pair of pliers, electrical tape, and a Phillips screwdriver.
First, pull all four knobs off. Notice the electrical tape on the right side knob- it increases your grip if you're having a hard time removing them.
Remove the 2 screws shown in next photo.
Grab the top corners as shown with your thumb & forefinger, and ease them out. As you do this, you'll hear the face panel snapping out, because it's still held in by the button assemblies.
Wrap the jaw tips of your pliers with electrical tape to grip the bulb, and pull it straight out.
After you replace the bulb, snap the face panel back in by grabbing the knob shafts while pushing the panel in. If it doesn't go in all the way, don't freak- replacing the 2 screws will draw the panel further into the correct position against the unit.
Push your knobs back in, and enjoy your newly visible in the dark climate control