One does not till perennial weeds. They have persistent roots which, after they are chopped into pieces, yield many more weeds than before.

You can get rid of them organically but it won't be fast. Dig out what you can then, every time you see a new sprout remove it. The goal is to starve the roots.

Many of those "weeds" are also wild flowers. The most environmentally friendly, organic, method of "weed" suppression is to deprive them of access to the sunlight they need to grow, cover them with a good thick mulch, or newspaper, cardboard, etc. and cover that with something, vegetative so it also contributes to the health of the soil, to hold the paper in place and hide it.
Tilling will quickly mix the present growth with the soil but will also turn up many "weed" seeds that will compete with what you plant there. Some people will till now and then again when new growth appears and then again when new growth appears until no new growth appears.
There is no environmentally friendly spray material that kills plants.

You could spray with horticultural vinegar to kill the annual weeds, then dig up the perennial weeds.

I think that kimmsr's suggestion of layering cardboard then mulch is the least manual intensive yet organic method you've got. It presupposes that you are not in a hurry to plant in the bed, as that cardboard is going to be there for a couple of months at least.

what is horticultural vinegar and where do I get some?

Horticultural vinegar is less diluted than household vinegar; I think it's diluted to 20% vs. 5%. I got mine from AM Leonard, but other places have it - lots of organic places carry it. The shipping on liquids can be a beast, though. Here's the AM Leonard page, to get an idea of what it goes for.

Here is a link that might be useful: AM Leonard's vinegar