I always say if you want to run faster, you need to spend more time running faster.
In general though, your workouts for 5k's should consist of both runs that are longer than the race distance although at a slower pace, (I like to see people in shorter distances going at least twice the race distance or at least 6 miles for 5k) on a regular basis. But you also need to do interval and hill workouts at a pace faster than your race pace. Probably the best interval distance for 5k's is quarter miles. So, the interval workout would consist of 10-12 quarter miles at whatever pace you could maintain for all 10-12 of them. Hopefully this would be up to 20% faster than your race pace. Put maybe a minute rest between each one. If you are new at it, you probably need more rest, but decrease the rest as the weeks go by. You can do the same thing with hill repeats, but maybe a little shorter and maybe a few more of them. Use the downhill jog as your recovery. For variety and to enhance your conditioning, you can add other types of speed work, distances, rest and hills etc. Probably twice a week is sufficient with a little lighter day before and a recovery day after if you need it. And with any workout of higher intensity, you will want to make sure you properly warm up and warm down.
So with all of that, you have comfortably exceeded your race distance and your race pace and should over time be able to put the pieces of running faster that you have practiced into a sustained effort that, while not as fast as your intervals, is faster than you have been racing.
It is also helpful to be mindful of your form to make sure you are running as efficiently as possible.
You could also add some weight workouts and cross training. A high intensity sport like soccer will approximate an interval workout, and swimming will help you work on breath control with no impact etc.
I would recommend that you work out schedule that will end with you being ready to run a fast time on a 5k. I will assume that you are in good running shape already and that you have been running at least 5 miles a day at least 6 days a week with one day rest. Give your self a 12 week time frame to achieve the desire fitness for the 5k. Out of the 6 days 2 should do an LSD run on flat ground at a nice even tempo not too fast to not slow, 2 should be LSD run with hills at a good tempo as well, easy hills. 1 should be hill repetitions of at least 200 yards/meters at a tempo pace 10/12x, with easy down hill as rest period, 1 day should be repetitions on flat ground of one milex2, half milex4, quarter milex6 timed- the rest period for these reps will be the time that you ran each. Make sure you warm up and stretch properly. For a 5k you should be running 35/40 miles a week.
For maximum improvement in your 5K time, follow a a 28-week periodized training plan customized for you after your next break from running. Aerobic improvements come from physical/structural changes which take time.
The quickest improvements typically come from changes to your breathing, pacing, race strategy & tactics.