I wouldn't use weedkiller either but if you are not intending to be an organic grower then you could do that.
The main point of raised beds is to do away with the need for digging (as well as creating a decent thickness of soil on your chalky ground). You will need to dig to remove the weed roots, but not to turn in manure - you have a healthy population of earthworms and this is what they do, turning in rotting plant matter and drawing it deep into the soil, whilst aerating the soil too. Plants need nutrients and oxygen and the earthworms get both to where they are needed, down at the roots. Earthworms also improve the natural structure of the soil by making it crumbly (soil passes through their bodies and picks up mucus on its way and it is this which helps with the crumbly structure) So use a mulch, once you have got rid of the weeds. By mulch I mean a thick layer of well-rotted farmyard manure or compost. Use compost on beds where you intend to sow carrots and other roots which will fork (ie the roots will divide into several smaller roots, which are not so easy to prepare, although just as edible) in manure, and use a thick mulch of FYM for greedy feeders. The mulch will also protect the soil surface from erosion by heavy rainfall.
When you are digging out the weeds, keep your eyes open for slugs and slug eggs (like tiny clear marbles, in groups of 20-30), especially against the wood edging. These are likely to have increased in numbers since the last gardener worked your beds so could do a lot of damage to your new seedlings.