as others have said generally you go to the bullet manufacturer's site and they will tell you. If you have an odd bullet you can use math (or JMB's website which I couldn't get on right now???server down?). It is based on the known tradjectory or known velocity's at two known distances. Don't put too much faith on published BC's -- they are a tool to find your true tradjectory. you must test each in the field since they are often slightly off in real world condition.
The computer program is McDrag. It's a empirical model developed by Robert McCoy at Aberdene Proving Grounds.(US Army), based on bullet shape and weight only. To use it you need precise measurements, but that can be done with a caliper if you're careful. It has a sister program McGyro which calculates bullet stability, but that program needs to know the internal mass distribution.
Realize that unless your bullet matches the shape of the particular drag model (like G1 or G7) it's only really valid at one velocity. The differences with bullets shape are greatest around Mach 1 and differ considerably from Mach 0.9 to mach about Mach 1.5. Above or below that a measured BC is generally close enough.
Method 1. Measure the muzzle velocity with a chronograph and the drop of several shots fired at at least three distances. Curve fit using a ballistics program. The method is affected by vertical difference in three shots for any reason including aim, wind and bullet yaw. Still its easy and give s good ballpark BC if the shots are made a reasonable ranges ( like 100, 400, 600 yards). Use a tall target and zero it on an aiming point near the top at 100, then use the same point of aim (no scope adjustment on the more distant targets and measure the two "greater" drops. It's the easiest method with the least equipment Chronograph each shot several feet from the muzzle and use that point to measure the target distances, not the rifle muzzle. Good velocity measurements are important becasue what you're trying to measure is the difference betwee the actual drop resuting from the bullet slowing down from air drag. and the drop due only to gravity. If you're ammo has high SDs (velocity variation) from shot to shot) then your BC measurements will be poor unless you apply the velocity of each shot to each fired bullet rather than using averages. In theory with four targets at different distances you could determine BCs withhout the velocity measurement, but in practice small measurement errors would give large errors in the BC..
Method 2. Measure velocity at two points with the same bullet and a known distance. Fit with a ballistics program. Requires two chronographs. The further one seems to want to get shot. Use a steel plate aperture in front of the far chronograph. The JBM site has an on-line version of both programs which work nicely.
Method 3. Measure one velocity at one point and the time of flight over a known distance. An acoustic transducer bonded to a steel target can give accurate time of impact relative to the bullet passing the chronograph. Use a ballistic computer to fine a match. Downside- requires the transducer and stringing wires plus a customized chronograph.
Methods 2 and 3 are the most accurate, but they all rely on accurate chronograph measurements. There are pitfalls in all of these methods. If the bullet is yawing (not stable) it will give low BCs. Thats not bad if you use the same rifle you'll be shooting with as it gives realistic BCs. Also realize that BCs are calculated for standard air density and have to be corrected for the air density where you do the test. Air density is a function of temperature, humidity, and absolute pressure. Absolute pressure is a function of normal barometric pressure and altitude. Again, most ballistic computers can help sort all that out. All of the methods are affected by head or tail winds. In all methods the distances need to be accurate. Use a laser rangefinder.