While I am not sure how to go about strength training, and I know that it is one of the most important aspects to develop speed, and that Steven knows his stuff, for running a sub-25 5-K, I can give you my cross country summer training schedule. I am in high-school, so this is for 5-K raceshttps://docs.google.com/document/d/1OUn5V6bV2I6OYXY0jZDLeKpYEVZHY60pjk4oggI2ZKI/edit
If you want to start this up to get a better 5-K time, you should keep up with your strength raining and slowly add in this routine. It is important to note that the first two or three weeks should be mostly LDS (oh no!) just to get used to running for a decent period of time. If you feel comfortable with your running experience (i.e. already run fairly regularly), by all means begin adding workouts from the beginning.
After around two weeks, if you are following this type of training plan, you should begin to incorporate faster, or hilly-er runs in as kind of light workouts. A "tempo run" is a run where you are picking up speed from the very beginning for the run, at no point after the first two minutes should it feel comfortable. You should be running fast, if were talking times, if should feel a little slower than your race pace, by about 1-2 minutes. A hilly run is pretty self explanatory, you run up hills fast and never stop moving. Jog down the hills for recovery before attacking them again. Make sure that you really focus on nearly sprinting up the hills, for about 8-12 hills.
These more strenuous runs should be done once your running four to five times a week consistently, and should never be done more than three times a week. other days should be for strength training and recovery. recovery should be at a short run at LDS pace to flush your body and reduce DOMS. Lift heavy on these days.
If you want, you can also do 1 "long run" per week. This should be 10-15 minutes longer than the normal amount of time you run, and should be fairly slow. I don't really like these and never found them helpful, but many of my team-mates have made significant improvements to their times when they incorporated these into their training. Meh.
Following these instructions and continuing this type of program for about three weeks after I finished the basic summer plan, (we didn't add much time of distance, we just added tempo runs and workouts) I ran my first 5-K last year at 22:35. You might have different results, but for the times you are going for, this is a good plan.
If you seriously want to run a good 5-K, you need to get in a race. Having people around you to push you forward helps immeasurably to focus you on the race. You don't need to make it competitive if you don't want to. You can find someone to run with, and try to stick with each other to whole race. If they are obviously faster or slower than you, don't even bother as you will either gas yourself or hold yourself back. Training with a friend also make running much more enjoyable.
But keep in mind that this kind of running will almost certainly hold back your strength goals. I do it because its fun, but I understand that I could be much better at parkour and much stronger if I didn't run longer distances. Mods, please correct me if I have any wrong info about training for this!