First, improve your starting position at the line. Many football players overlook their starting position and continue to make technical errors that hurt their 40 times. After all, you won't be able to get off the line quickly if your initial positioning is all wrong.
Before you can do anything else, you need to figure out which foot should go in front. The best way to do this is to stand with both feet together and have a teammate push you lightly in the back. Whichever foot you move first should be the back leg in your starting position.
Place your front foot right behind the starting line to get your center of gravity (usually around the navel area for most players) as close to the finish as possible. Taller players may need to leave a couple of inches between the foot and the line to feel comfortable.
Bend your back leg so the knee is close to the ground, almost touching the starting line. Place the hand on the side of your back leg on the ground and hold your other hand near the hip in a "ready" position.
The hips should be slightly higher than shoulder-level, even though this probably will feel uncomfortable. Many players start with their feet too far back because it feels better, but this severely limits your explosiveness off the line. The ideal starting position should feel cramped, as though their body is recoiling before exploding at the starting gun.
Now that you've got the correct position at the line, it's time to work on the first step. A good first step is vital for a fast 40 yard dash time. Correcting the starting position usually also improves the first step because you're starting closer to the line. You want your first step to cover as much distance as possible without sacrificing the proper body angle.
One easy way to practice the correct stride length is to tape off a circle where you want your foot to land on the first step. Measure the length of your femur (big bone in the thigh) and use that length as the starting point for where you will place your tape circle. Practice taking off from the line a few times and adjust the location of the circle until you find the best spot.
If your step is too short, your back foot is probably too far back in your starting position. Don't step too far either or you won't be able to drive with your legs and generate power. Ideally, you want your leg to be striking the ground at a 45 degree angle, not straight up and down. Hitting the ground with your leg perpendicular wastes most of your explosive force pushing you up into the air instead of forward toward the finish line.
If you can, have a friend or teammate film a few of your practice runs. It's easy to check the details of your technique when you can slow things down or rewind the tape. It doesn't have to be a high-quality recording for this purpose. A cell phone that can capture a few seconds of video should be all you need.
Practice these first two parts of the 40-yard dash and you'll see your time improve without any extra workouts or running drills. Of course, you should still run and lift weights to get your absolute best time, but these quick adjustments will give you a great head start to running a faster 40-yard dash.