Hard work and muscles! Cut each tree/shrub down, get rid of them first, then start digging out the roots. Sorry - but there is no easy way round this. Once the hedge has gone, you'll have to dig in lots of compost and well-rotted manure to replace the nutrients that the conifers will have taken out of the soil, before you can plant anything else there.

If it's a big hedge, consider hiring a small digger (mechanical, not human!!) to remove it. This will cost money but save lots of time. Then have a huge bonfire!