Over the years many hobbyists have wondered over how to breed Koi fish. The truth of it is that in most cases Koi will breed in your pond all by themselves without much help. There are many factors which one must consider when keeping Koi with the intention of breeding and even if there is no intention for breeding.
The first thing the serious hobbyist will notice is that one or more of his Koi had become quite fat towards the end of the winter months, in the event that he did not over feed during the colder months this would indicate a pregnant female Koi. The next noticeable action in the pond is the males, which would be slimmer and more streamlined with wider larger fins, would be chasing the females around or hovering behind them. In general however the behaviour of the males tend to become quite aggressive constantly pushing and shoving at the poor pregnant girl they may start harming her if there are to many males in the pond. You should at this point remove any pond ornaments or equipment with hard sharp edges (sharp edges should never be there!) to prevent the female from getting pushed in and being seriously hurt. You should also consider putting a net over the pond as the females may be inclined to jump at this time.
Generally Koi spawn from mid spring to mid summer when the water has warmed up nicely. If you have been keeping an eye on the fish and noticed the above mentioned behaviour it is advisable to check on the pond every morning before work and straight after work. If you notice foam and bubbles or an oily type substrate floating on the pond it will indicate that spawning has taken place, you will also probably notice a strong ammonia smell. The biggest danger for the hobbyist is the ammonia, if your pond is on the small side or your filtration is not up to scratch your fish may be in mortal danger from ammonia poisoning! You should test the water straight away and make sure the ammonia levels is at an acceptable level, if it is too high you will have to do a water change and use a de-chlorination product.
You will of course want to look for the eggs; normally you will find them on the sides of the pond if there is some algal growth or if you have stringy algae or plants in the pond. If you intended for the spawning to take place you could hang spawning brushes on the side of the pond. Lets say you have no intention of having the eggs hatch or raising the young, you could immediately scrape at the pond walls where you see the eggs either netting them out or pushing them to the bottom drain, next you should do a large water change, there is no need for you to take a risk of any ammonia poisoning if you have no intention for getting some benefit from the process.
If indeed it was your intention to breed with your Koi, have a look at some of the other articles regarding spawning and raising fry that I have.