It is not possible to sex koi which are less than about 10" long as they are sexually immature, and it can still be difficult when they mature as there will be the odd skinny females as well as some fat males. Generally speaking though, female koi are plump (due to egg filled ovaries) with smaller pectoral fins and males are streamlined and more torpedo shaped. Males also develop breeding tubercles on the head and pectoral fins in the breeding season. These breeding tubercles are tiny raised spots and are difficult to see, they are occasionally mistaken for white spot (ichthyopthirius).
Left on their own, koi (if they are more than 10-12") will naturally spawn in the early summer. A water temperature of about 20C (68F) is ideal and the koi will spawn 'en masse' (known as flock spawning). This will produce healthy offspring but generally poor colors, so if you want to breed good quality fry you should choose the parents and put them in a separate pond (a kids paddling pool will do.) You can put 2 or 3 males in with one female.
If left in the pond most of the eggs will be eaten, and any fry that do hatch will usually be eaten too. If you want to save some fry, you will have to move them to a tank or pond with no adult koi. It is very difficult to collect eggs that have been randomly distributed in the pond so it is a good idea to use spawning brushes which the koi will spawn on. Place the brushes with the eggs on in the pond/tank you are going to rear the fry in and add 0.2mg per litre malachite green solution to stop fungus infection. It is preferable to add an air stone as the developing eggs need plenty of oxygen. If you are going to filter the tank make sure the young koi will not be sucked into the pump, you could put a fine net around it, if you are not going to filter the water, you must do regular partial water changes (about 20% of the water).
The day before hatching the eyes in the egg will have a shine to them, soon the koi will begin to wriggle and in a few hours break out of the egg. At temperatures of about 20C (68F) it will take 3 or 4 days for the eggs to hatch. If the temperature is slightly warmer, the eggs will hatch slightly sooner, but be careful not to overheat or rush the process.
It is very important that the fry have plenty of oxygen at this stage so an air stone is a good idea. After hatching the fry will stick themselves to the sides of the tank with a sticky pad on their heads for 2 to 3 days, then they will swim up to the surface and gulp some air which is forced into the swim bladder and enables them to swim freely. This is the time to start feeding them. Hard boiled egg yolk is a good food for the first day or so, it has little dietary value but will increase the size of their stomachs. Newly hatched brine shrimp are also good, feed these for about a week and then the fry will be ready to eat special fry food (very fine powdered pellets). When they get to about half an inch feed them tiny pellets and increase the size of the pellets as they grow. Remember not to overfeed and check water quality regularly, fry are very susceptible to high ammonia levels and it is easy to lose them all if care isn't taken.
Even if all the proper safety considerations are taken, you may not have success for a few years. BPP wishes you great success in breeding, but recommend patience in re-creating a quality koi.