For your Tektros you would need "Mineral Oil" fluid. It's NOT THE SAME a DOT Fluids, don't mix them. Urban legend says that would make them go kaboom.
To replace the fluid:
- Hang your brake, in a way that the caliper is at the bottom and the lever is leveled at the top. Work while is on the bike stading up.
- Put a tube at the the caliper, remove the rubber cap that you can see in the picture you uploaded.
- Put the other end into a bottle with a little bit of mineral oil. This way it would suck up mineral oil instead of air.
- Put something beteween the pistons, like a rotor, or a pad separator. This is important because you will need to pump the lever a few times and you don't want your pistons moving around.
- Open the top and have you oil ready to pour.
- Hold the brake lever, softly, it need to bit you separator.
- Open a little the caliper end, where you attached the tube. You'll see and feel the oil flowing.
- Top up the oil in the reservoir and do it again a few times. You will see a change in color of the fluid going out, and you should not see any bubble.
- Repeat till you are happy with it.
- On your last pump, close the caliper end while the lever is pressed. This way it won't suck any air, or old oil, from the tube or bottle.
- Top up the fluid and close the reservoir.
This's was the way I did it on brakes like this, always worked. If you still have bubble in your system. With everything closed, hang it fully upside down and vibrate it a little. So the bubbles will go to the caliper. Then put a tube a the caliper end, press a little the lever, open up the caliper, don't depress the lever and close the caliper again. Air in the system should be trapped in the tube mixed with a little bit of oil. Now you should be almost without bubbles.
PS: Brake fluids are nasty, use gloves and cover everything where you are working.