You do not want to create a bubble of fluid directly under the skin. Do not attempt to tear the skin away from the fatty layer, you will just end up bruised.
Copied from elsewhere:
Place your thumb and forefinger on either side of the injection site about two inches apart. Pinch up the skin.
Insert the needle quickly into the pinched skin between your thumb and forefinger. The needle should enter at a 45-degree angle (slight angle) or 90 degree angle (straight up and down).
Release the skin and hold the syringe in place.
Pull back slightly on the plunger. If blood appears, the needle has entered a blood vessel. Remove the needle and begin again at a different injection site.
If no blood appears, hold the syringe in place with one hand and push the plunger in slowly with the other hand until all the medication has been injected.
As you remove the needle from the insertion site, place an alcohol swab over the injection site.
Gently press the alcohol swab against the injection site for a few seconds.
Do not recap the needle. Dispose of the needle and syringe in the disposable container.