Here is my Quick & Dirty Method for teaching "Greasing the Groove." The basic idea behind "greasing the groove" is to get better a specific movement, usually for some kind of test (such as a pullup test, pushup test, or situp test). There are other benefits, but the goal is improving your maximum number of good reps so you can excel at the movement when called on to do it.
The basic idea is to do lots of easy reps, regularly. You do half of your maximum one-set maximum number of reps of that movement throughout the day. By doing so you improve your ability to do the exercise and do it more efficiently, and thus you can do more in one long set.
While there are (very good) strict ways to do this, I find a looser method works just fine.
Replicate the Test's Technique. Whatever you want to improve, do it the way you'll be tested. Paused pushups? Do them paused. Dead-hang chins? Do them dead-hang. Touch-and-go bench? Train it touch-and-go. It's test practice, so practice the test. Follow the rubric.
Half your current max. Not "best ever" or "goal" but "I could do this right now." If you could drop and do 12 now before you get tired and/or sloppy, do sets of 6. The only exception is if you can get only 1. Then do singles, with plenty of rest between them. Be honest and only count the good reps. If you can't be honest or can't self-evaluate, have someone else do it and tell you your actual max number of good reps. Halve that and do sets of that.
No hard reps. No grinding, no fatigue, no bad reps. NONE. No wiggling up in a pullup, bridging in a bench press, kipping in a pullup, or foot-down assists in a one-legged squat. Good reps. Easy reps. You're training in good technique and skill. It's practice, not strength training. Just get in easy, perfect practice.
Many sets, by feel. Don't worry about how many sets you get in during the day, as long as you get in at least 3 of them. Anything past that is bonus, and even 2 sets are okay if that's all you can squeeze in. But in general, try to get in as many good sets as you can. If you feel fatigue after a set, wait longer before you do another set.
Be opportunistic with these - I used to drop down and knock off a quick set of pushups waiting for people at job sites. The usual pullup recommendation is to knock of a set every time you pass the pullup bar (or at least, knock off a rep or two.)
One day off a week. Once a week, just take the day off. If you can't stand that, do one set and stop there.
One test day a week. The day after your day off, warm up a little and then do a max set (in whatever form you need to complete the test you're prepping for.) Use this to set your new max. If it's more, perfect. If it's less, scale back a little (you might have overdone it.)
Have fun with it, and remember - practice. You're just doing some easy practice of the movement sporadically throughout the day, getting better. Don't make it a chore or a grind, but make it something you do often. It'll work, sooner rather than later, if you stick to the guidelines above. If you stall out, take a step back and evaluate. You might need to reset and start over with fewer reps per set, or you might be ready for a stricter program. But when I'm asked how to get more pullups or pushups, this is my five-minute spiel on getting more of them.
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