How to Learn to Cast a Fly... While Sitting in a Chair
The first and most important tip is to learn to cast on a dry surface. Once this has been mastered it will be easier to cast on a water surface.
I taught myself to cast from a book and practiced on a soccer pitch close to my home. And after some time I was able to cast reasonably well. Of course I had to put up with the many quips as to "How many fish have you caught, mister?"
However it was only after attending a fly fishing clinic held, some 25 years ago, by the master fly-fisher, Mel Krieger, that my casting improved leaps and bounds.
Mel taught the class to do all sorts of casts from the roll cast to double haul techniques. But the most important lesson went something like the following and it applies to all forms of fly casting.
It was Mel who showed me how to learn casting while sitting in an armchair watching TV and holding a short stick although you don't even need the stick.
Mel Krieger is the author of a great book called "The Essence of Flycasting" ... well worth buying if you want to learn about the wide range of casting techniques all of which require firm control of the rod handle.

The Important Points in Learning to Cast a Fly Well
Imagine you had a hammer in your hand. Now imagine that you had to knock a nail into a 3 inch thick vertical wooden panel.
Think seriously about how you do this ... (1) you do not swing your arm through a wide arc of say 90 degrees or more (2) you do not gently hit the nail with the flick of a wrist either. (3) You do aim to punch (this is important word) the head of the nail with enough "Oomph" that the nail penetrates the wood without going all the way in. (4) Your hand comes to a sudden and abrupt stop just in front of your eye and not close to your waist. (5) You then bring the hammer back through a small arc of about 30 degrees only or maybe to a point just behind your ear. You then hit the nail with "Oomph" again.
This is how you handle the forward cast using not a hammer but a rod giving enough "Oomph" sufficient to propel the fly and line forward.
That's how you do a forward cast ... no limp wrist, rather a firm determined movement of a stiff arm and wrist rocking through a 30 degree arc only. When you become proficient you will tend to use the wrist a bit more but not in the early stages of learning to cast well
You do the backward cast by imagining the nail you need to knock into the wooden panel is directly behind you. The full cast therefore takes place in a small arc of some 60 degrees only.
The Armchair Fly Caster
Now that you know about nails and the need to "Oomph" the rod with a stiff arm take a stick about 2 feet long, sit back in the armchair and practice that action of hitting the nail into the imaginary front and back wall. Make a very deliberate and even exaggerated stopping of your arm in the forward and backward 60 degrees arc movements motion just as would be the case when coming into contact with the imaginary nail. Wait about a second after making the back cast before starting the forward cast with the stick.
If you transfer this action to the handle of a fly rod used to cast on the lawn you will quickly become an accomplished caster