The first requirement for successful execution of the screw shot (also known as the draw shot) is to aim at the very bottom of the cueball not by lifting the butt end of the cue into the air but by lowering the whole of the cue parallel to the bed of the table. A good tip here is to flatten your bridge hand. Whereas in the standard position the cue rests around about 2" above the cloth (at the point it crosses your thumb-forefinger rest), by flattening your bridge you should be able to lower this.
Here's a quick visualisation technique to help you aim at the correct point on the cue ball. This applies not only to the screw back shot but to any shot you will ever play. Picture the cue ball being split into layers. The very bottom of the cue ball is number 1, the centre is number 5 and the very top is number 10. This is a technique applied by the very best of players as it helps to engage the brain into a positive thought process. By successfully implementing this into your own game, you will find that you have a clearer understanding about how to execute every shot you play because you can apply a number to the position on the cue ball (e.g. stun shot = 5; stun run-through = 6-7; deep screw = 1-2). Please consider that you must also consider the power in which the shot will be played and the timing of the strike as these two factors obviously affects the action of the cue ball after contact with the cue tip. The same rules apply to side spin or 'english'.
Although in my opinion, the way you grip the cue is not so important, it still warrants discussion because the grip itself has vital importance during the follow through phase of the cue action. Your grip on the cue should be guided by your thumb and forefinger. The other three fingers do not need to play any part in the cue action. They simply 'do as they please' during backswing and follow through. Movement of these fingers should be unconscious. With the screw back shot, it is important that your grip is loose enough to allow you to pull the cue back as far as possible - if it is too tight, you'll simply not be able to pull the cue back far enough.
Here's the important part regarding follow through - to achieve maximum screw back you'll need to learn to follow through that extra little bit further than you may already do and to achieve this you may find that you'll need to let your elbow drop. Watch snooker player Ronnie O Sullivan - notice how his elbow drops when he plays any shot with high power. This is because his grip has followed through maximally, i.e. past the point where it usually stops at - when it hits your chest. If you have your elbow locked you'll find it difficult to follow through effectively for the screw shot. Try pushing your cue through another 2-3 inches after you feel you have reached full follow through and you'll find there's a little bit more to go by letting your elbow drop as you drive the cue through - at this point make sure that you don't 'release'. What I mean here is that you must keep your thumb and forefinger locked because they guide the cue and any release can cause the cue to throw off-line. If you can implement these little 'maneuvers' into your game you'll notice your timing will improve as well because it will stop you from jerking the cue at the end of your follow through.
Many players have a good cue action but seem to be afraid of cueing too low on the white in fear of the dreaded miscue. Well, most of you will know that you can literally rest the tip of the cue on the cloth directly behind the white ball (watch snooker player Jimmy White for example) because a good cue action follows a pendulum like plane and at the moment of tip-cueball impact the cue is actually on the up.
To summarise and to put things into perspective....the reason for a long backswing is simply because of the point I mentioned above regarding the cue action being pendulum like. Visualise a swinging pendulum and how would you describe it....simply that it swings a certain distance in one direction then the same distance in the other direction. Now think back to the cue action....it's simply inefficient to expect to bring the cue back 6" and to follow through 12" - that's not pendulum like is it!? Aim to bring the cue back by around 8-10" to your thumb-forefinger rest (this should tell you where to place your bridge hand in relation to the cueball). When you are ready to follow through, you should attempt to push the cue through the cueball with a fluent motion by 8-10" also. This is arguably the most difficult part of the shot. If you can achieve this or even get somewhere near then you should be well on your way to improving your execution of one of the most difficult shots in the game to execute well, the deep screw.