Killing off all of the insects in your garden is not a desired result, as any healthy ecosystem requires an abundance of beneficial insects, microbes, and fungi, both in the soil and on the plants themselves. Introducing other predatory insects such as ladybirds is benficial. However when faced with an infestation, using a homemade insecticide is a safer option than the more dangerous toxic shop bought products.
All you need is a bulb of garlic, cooking oil, water and some washing-up liquid. Crush all the cloves of garlic and put them to seep over night in 2 tablespoons of oil. The next day sieve out the garlic and add the oil to just under a litre of warm water, with a touch of washing up liquid into a spray bottle.
As well as removing aphids this is a good spray to use as a slug deterant too.