About ten years ago I was fortunate enough to attend a clinic that Vern Gambetta spoke at. At the time I had no idea who Vern was. Lucky for me the person I was with did. I learned a great deal in that hour, some of which I still incorporate into my daily training. One of those was his explanation on how to do a proper push-up. To often I see people doing improper push-ups, so I figured I pass along what I learned from Mr. Gambetta.
The key is hand placement, try to keep your hands as close to your body as you can.
Start in a Prone Handstand.
Remembering not to compromise your plank pose on the way down
Or on the way back up.
And always strive for full range of motion!