The textbook answer is to skim the top layer of turf off; with about and inch of so of soil. Find an unused corner of your garden and stack the turves, upside down and let them rot down as they will make a pile of fine soil eventually. You can then, dig the remaining soil over, incorporating some well-rotted manure, compost or soil improver.

Agree 100% with Jimmy.

Either way you will find that once exposed, hundreds upon hundreds of dormant weed seeds will start to germinate in the turned over soil. So if you have a weed preventer available at the GC apply it to the newly exposed surface and to the turned over turf which will be as full and green as the side you flipped under in several weeks if you just leave things the way they are. What you don't want to be saying to yourself is, "Where did all this come from?"

Or if you don't like using too many chemicals you could just leave the soil bare until planting time in the autumn, in the meantime hoeing off all the weeds that appear, and digging out anything substantial such as dandelions etc. You could cheer it up in the meantime with border of bedding plants if all that bare soil depresses you. I don't think you can still get pre-emergent weedkiller int he UK.

If you have skimmed that top layer of soil off with the turves, then their shouldn't be any weed seeds present. The only ones you will have to deal with will be perennial weeds such as docks and dandylions which you can deal with by hand. You can deep dig by making a trench at one end of the plot and turning the soil upsidedown into the trench as you go. You can also line the trench with compost or whatever to have.

If I was going to do that I'd bury the turfs upside down at the bottom of the trench as you go. Adds compost (eventually!) and saves carting them away to rot and back again to spread when rotted. Our garden was all grass when we came and I've done this for every bed we've made.