I have a somewhat different take on this. Pushups are basically a pectoral (chest muscle) exercise; if you want to do better pushups, then you need to train your pecs, and pushups aren't themselves the best way to do this because you're limited to your own body weight. There are far more effective pec exercises; the catch is that you need equipment—a power rack and a dipping stand—that you're going to have to belong to a gym to get access to, probably. But if you can get access to these, the following are top exercises for pecs (and will make it possible for you to knock out as many pushups as you like, at pretty much any pace):
Weighted Dips: get a chain weight belt (any barbell/weight-training store will have them, around $25 or so) and wrap the chain around, say, a 50lb dumbbell. Get in the dipping stand, face inward, with your hands gripping the arms of the stand on either side of you (not in front or behind), push yourself up into your strongest range position and do short reps, quickly. Do not do a full range of motion; this does not help at all. The reason is that the degree of muscle growth in response to stress is strictly conditioned by the overload, and you cannot carry anything like as great a load if you move out of your strongest range. Keep your reps short and fast, so that you cover the same amount of distance in a given unit of time that you would doing a slower `full range' rep; that way, your workload will be much greater (because you're carrying way more weight over that distance) and the power (work per unit time) you're generating will also be much greater (because you're moving that much greater weight the same distance in the same timespan). My `cruising' weight for this exercise is a 120lb dumbbell over a very short range of motion, and believe me, this gives your pecs a tremendous workout! Give yourself a long recovery on this exercise—start off with once a week, then once every two weeks. When you start lifting three figures on top of your own body weight, you have to allow a lot of recovery time in between sessions or you'll lose condition; muscle growth only happens after recovery, and recovery from high intensity weight training isn't like recovering from most common training regimes with moderate weights and high reps.
Short-range Bench Press: By restricting your benches to short ranges of motion in a power rack, you can add around 100lbs to your current bench right off the back. Set up the power rack so the weight rests are about an inch above the pins, and so that when you lift the weight off the pins, you get at most three inches or so of extension. The deal is the same as with weighted dips—short, rapid reps at increasingly heavy weights. Over the course of about three years I worked my way up to a max of 405lb short range benches, and I'm pretty much the classic hardgaining ectomorph. If it worked for me it'll work for anyone!
If you do these exercises, with proper recovery, for a few months, you will find that you can get down on the floor and blast out a hundred pushups any time you like, any pace you like. They'll seem almost trivial, and they are, compared with what you have to do on the weighted dips and benches. Your chest muscles will get dramatically stronger. Just keep your workouts short and very high intensity, with lots of rest in between sessions. People who've used this Mike Mentzer/Power Factor style of weight training get very strong relatively quickly, but the down side is that the nature of the training requires you to be in a kind of aggressive mind set every workout. If you're doing things right, you should be prepared to increase your weights and maybe your reps at the same time or less than the previous workout. It's easy to become intimidated at the thought of what tomorrow's workout routine will consist of; you just have to resist that line of thinking and tell yourself that the iron is an opponent you can defeat....
