Problems and Limitations
The Flash is a supernatural (and made-up) entity, which cannot be replicated by science in any way, shape, or form. However, pushing people to simply run faster is, of course, possible, depending on what you're trying to accomplish, and price you're willing to pay (or rather the price your test subjects are willing to pay).
Consider that the current world men's record for sprinting 100 meters (held by Bolt), is essentially 23.35 miles/hr. That is the fastest that a talented, and highly trained human being has ever been ever to run (that we know of).
He did this at the peak of his physical performance, although not on any performance enhancing drugs (we hope). And so, right off the bat, you're opening the door to potentially having him run faster if he did partake of some of those drugs.
However, everything has consequences. Running that fast will take a heavy toll on most people. They will most likely tear muscles and ligaments achieving higher speeds, because the human body has certain limitations. And they will still come nowhere near 60 mph. In fact, most would be lucky to reach Bolt's speeds while not on drugs.
Also consider that many professional athletes develop a wide range of injuries related to constantly training, and performing to the limits of their physical ability. The people enhanced in this manner would likely suffer terrible injury much, much sooner in their "careers". Most likely such an enhanced person would maybe have a career measured in a few tens of kilometers, not years.
What you might try is a combination of drugs which enhance blood oxygenation (which is what Armstrong was taking to stay at the top of the bicycling competitive world), and maybe replacing people's legs with prostheses along the lines of the Blade Runner.
Alternatively, the world's militaries are currently developing powered exoskeletons. These things are primarily meant to enable troops to carry large weights. They're currently nowhere near what we see depicted in movies, or video games, however in a few short years they may well get there. These things may very well allow a human being to move very freakin' fast. Maybe not sustain a 60 mph sprint, but certainly move at Bolt's speeds for short periods of time without having to be anywhere near as highly trained as him.