For running faster 5K with 3 workouts a week, I'd do something like:
1 long run - 5 miles or so at an easy pace
1 run with speed intervals. Depending on your fitness level, you can walk in between intervals, or jog slowly to recover. Start out with 400 meter repeats and work up to 800 meter repeats (fewer repeats as the distance increases). I prefer to do these on the treadmill, but YMMV.
1 tempo run - probably about 2 miles, plus warmup and cooldown. This is two miles "comfortably hard" i.e. tough, but not feeling like you're going to puke.
You have room to throw in a 4th easy run with this schedule if you have the time for it.
Other things that will help: strength and cross training - deadlifts, squats and lunges to strengthen your legs, cross training with rowing, jumping rope, cycling, box jumps, stair climbing, kettlebell swings etc at least 1x per week.
Check your running form. Make sure your cadence is quick, be sure not to "plod" even on your easier runs - i.e. shorten your stride length to go slower, rather than reducing your cadence. Make sure you have good posture, keeping your shoulders back and running from your core. Don't collapse at the waist, this just burdens your legs as they get more tired. Swing your arms forward and back from your shoulder, not side to side or anything else - they should be pumping you forward, and your arm swing can influence your cadence as well. Pay attention to that.
I didn't have a 5K time last year, but my best 10K was 56:30 and I just ran a 23:10 5K last weekend after following the above advice over the winter, so it definitely works.
Voluntary Discomfort is the secret cornerstone of strength. We build our whole lives around increasing comfort and avoiding discomfort, and yet by doing so we are drinking a can of Weakness Tonic with every morning’s breakfast. ~Mr. Money Mustache
5K PR: 23:40
10K PR: 48:57
HM PR: 1:59:37