Here is something I do when I'm in your situation. Shallow dive, on his six and he starts to turn:
1. I really start observing at about 1.5 to 2.0 K out. At this point, I try to estimate his E state very carefully. If I'm gaining on him fast, I make my dive even shallower till I get about 1K out. If not, I continue with that dive. At about 1K, most guys will already start to pull. Here's where you will run into two type of pilots. The first will pull hard into you, trying to get his nose back around on you as fast as possible. The kind of pilot will pull a gentle turn at first and then tighten as you get closer. The latter of the two is the one I worry more about.
2. In the case of the first scenario, I will go to the vertical and see if he's willing to go with me (set up for a rope). If he does not, I either wing over or loop back over trying to get directly above him. In this position, I can get my lift vector in any direction really fast as all it takes is an A-Roll vs. turning with elevators. I then dive for the shot.
3. In the case of the second scenario.. well it gets tricky. I usually do a tight lag roll opposite his turn and keep lag pursuit. This will buy me more time and allow me to observe what he is doing. I want him to show me his hand. Some guys will break out of the turn and try to extend. Some guys will take it to the vertical and wait to see what I'm going to do. Some guys will continue a slow turn and wait for me to commit. Either way, there is a point of no return where I'm going to have to commit to the shot (at this moment, the danger for me is almost as great as the danger for him because depending on what happens, he can reverse very quickly and I'll be in trouble). For me, the trick is to set up a shot that has the most probability of landing hits. Again, in the end, I try to get myself directly above him where my speed may be equal to his but my total E is greater than his since I have altitude I can cash in for speed.
For me, the objective is to be as close as possible to him with greater energy before he initiates a defensive maneuver so that he will have less time to react and thus fewer options. This translates into a gun solution which has a higher probability of hits. When you are merging (I consider approaching him from the rear a merge also... after all, your two planes are merging) from 2+ K away, he has a lot of time to set up a defense and observe you. If the merge starts with (lead turn) at less than 1.5 or 1K, he has almost no time to react effectively and my guns solution will have a higher probability of hits.
Sorry if this isn't very clear but its my best attempt to describe what I would do. I guess a film would be better.
BTW, the most dangerous defenders are the guys who let you in close, present what looks like an easy kill, make you commit to a shot and then pull a vertical maneuver taking you completely out of phase, then getting his guns back around on you fast for a shot. I've been killed like this many times. If you encounter one of these guys, you know you just fought someone who has great SA, great ability to judge E state, great gunnery and great timing. Hate those guys