If your goal is simply to achieve the ability to do a bunch of push ups, you should attack it with volume (in other words, you gotta do a lot of push ups to do a lotta push ups).
Spread out the load throughout the day. Here's how I would take it if my max rep range was 5:
First, make sure that the five you can do are strict push ups with proper form (as Aizul noted). Without form, you're wasting your time. So test yourself accordingly and find your max; stop before you fail (grind out the last one if you need to, but don't stop half way and face plant).
Every day following, do 2 super strict, slow-and controlled push ups 4-5 times throughout the course of the day. Do this for a week, then retest your max.
- If your max is still 5, don't worry about it; repeat the program the next week. You'll eventually get to 6 or 7; when you do add an additional rep to your daily sets. For example: if you can do 6 reps consecutively, do 3 reps, 4-5 times/day. Retest the beginning of the following week.
Treat every daily session as push up practice, rather than a workout with pushups in it. What you're going for, especially in the early stages, is getting all the correct muscles to contract and inhibit in perfect harmony (while eventually strengthening the muscle fibers at play). At this point, it's closer to learning a new dance move than it is "exercising".
Once you get to the 20-25 range (I know it's arbitrary, but 20-25 is a solid foundation to build on...at my age, it would be enough to meet the minimum Army PFT push up score), start playing with different push up variations. Honestly, just Google "push up variations", and experiment. Use these new variations to program a path to your goals; whether they be strength, muscular endurance, or hypertrophy (muscle growth).
If you want to just keep on bumping up your max count (like...I don't know...100+) at this point, I recommend a program designed specifically for doing a lot of push ups. Stew Smith is a master in this domain.
He has a lot of different programs to help US military ace their fitness tests, but here's a relevant free one:
Stuck on Pushups? Try the Pushup Push Workout!
I tried the above myself (along with pull ups), and had very solid results (went from 51 to 64 in two weeks). I wouldn't recommend this without a strong foundation, and he says to only do this once, maybe twice a year; he's serious...it's taxing, but it's a great plateau killer. Here's a write up on my results if you're interested:
Over 20% Push Up and Pull Up Gains in 14 Days Using Stew Smith’s Program
Be patient and consistent and the results will come.