The real secret of getting a contact lens in your eye is this.

First, you have to get good at controlling your eyelids with your fingers. They're going to want to close. You have to be sure your fingers have control when that happens.

Be sure the fingers you use to hold the lids are dry and be sure the eyelids are dry. Rub them with a towel to be really sure. Top and bottom.

Practice keeping the lids open. Nothing works unless you can do this. I'm repeating this because it's so important.

Okay, now get the lens on the tip of your index finger. As with the eyelids, be sure the finger is also really dry. And make sure the lens is lightly on the tip of the finger. One point of contact. 

Then, holding the lids open, put the lens on the eye. Apply light pressure. Hold it there for a moment. Then lift the finger away from the eye, keeping the lids open. Pause. Be sure the lens isn't on your finger. If it's not on the finger, it's almost certainly on the eye. The lens sticks to the eye without much pressure.  

Take a breath or two and then let go of the lids. Blink a few times. If you have good vision in that eye, the lens is in. Otherwise look around on the eyelid. Or perhaps it fell off. Look around. But it probably didn't fall off, it's probably on your eye.

It took me a long time to figure this out. Now I get the lens installed first time every time. Amazing. Never thought I'd get there! Hope it helps.

PS: A follow-up piece, I forgot to mention an integral part of the process -- the mirror!