Whether you do a wide-grip push-up or a close-grip push-up, you're going to be targeting mostly the peck muscles, the interior deltoids, and the triceps. When you bring your hands closer together, it actually just means that your triceps are going to have to work a little bit more. You take away the larger muscle group aspect of it and you hone in on the smaller muscle groups, usually the weaker muscle groups which is why close-grip push-ups are sometimes more difficult to perform.
You're going to set yourself up into a plank with your hands not shoulder width apart like a traditional push-up, but right underneath. Again, close together, close-grip. Your fingertips are pointing straight ahead and you just want to establish your high plank position here, watching that your belly is not hanging out. Instead, you're pulling the abdominals in, engaging your core, not so much that you take yourself out of plank and get the weight away from where you want it which is to be in the upper body. I like to think of it as I just kind of on my tippy toes, putting as much weight into the arms as I can so I can most benefit from the upper body work in this exercise.
Elbows are going to point straight back. You're literally trying to squeeze your arms against your ribcage. You don't need to go any lower than 90 degrees. Then exhale, pull the abdominals in, and press the floor away. So again, squeezing your arms in, feeling them tight against your body, abdominals pulling in, head in line with your spine, exhale, bring it back up.
You're focusing on the elbow extension. Your triceps muscle is the muscle that extends the elbow. You want to get full extension so you most maximize your triceps work. Be careful if you happen to be hyper-extended that you don't push too far into that joint. Just a teeny micro bend is good for range of motion.