If you want to kill what is there, you can just cut it down and put the landscape cloth and mulch over it. It shouldn't be able to come up through landscape cloth, although a few weeds are capable of punching through it once in a great while, mostly perennial weedy grasses with thin, sharp, needle-like tips to the shoots.

Landscape cloth is a nightmare in a garden once the weeds start to grow and the cloth starts to deteriorate. Get rid of what you don't like, don't use landscape cloth, and mulch what you do put in.

I am with Laceyvail - don't use weedcloth in a garden! If you google weedcloth on many of the forums, you'll see lots of opinions that agree with this. Roots of woody plants from below and weed seeds that have blown in from above will root into the fabric and you will come to regret putting it in. It keeps mulch separated from the soil, so the soil doesn't get new organic matter and as the mulch breaks down and organic matter blows in you have new growing medium on top of the landscape fabric. It's an all around bad idea and the nightmare is when you decide to get rid of it . . . DH did this once many years ago and it both didn't work and was misery to remove a couple of years later.

It does have its uses - to keep drainage material from washing out from behind a dry-stacked stone retaining wall, to keep soil in a basket where you want to plant annuals, or under a walkway to separate layers of soil and drainage material, but you will come to regret ever using it where there are plants growing in a garden.

Instead of using landscape fabric, I mow down everything to ground level and sometimes will dig a few weeds that I know to be stubborn, put down heavy cardboard and then a heavy layer of medium fine mulch on top of that to a depth that settles to about 3 inches. That's enough to prevent most weeds from sprouting and to keep most of the already present weedy plants from coming up through. I let that sit for a few weeks to a year and then plant. If you are in a hurry to plant, though I can't imagine that this is a great time to plant in most areas of Colorado right now, you can use some type of contact weed killer (they range from gas flame weed torches to strong vinegar to Roundup.)

One more quick comment - since you are planning to redo the whole area, now would be a perfect time to add any soil supplements like more organic matter to help hold moisture and nutrients.

IMO the key to successful gardening is to do what needs to be done when it needs to be done the way it needs to be done.
That means not wimping out on the heavy labor :)
It's good for you- you can skip the gym that day.

Whether you are planning a new bed or rehabbing an old one the investment of time to do it right the first time is crucial. If you skip steps or do a half a**ed job you will be playing catchup and redoing things for the next ten years.

It's June already so too late for spring planting anyway.
Two choices:
Dig out all the plants.
Amend the soil.
Grab a chair, a cool drink, and study up on plants and and plan your design.
Plant in the fall.

Or if you really want a great bed:
Dig out all the plants.
Amend the soil.
Lay cardboard or layers of newspaper over the entire surface of the bed and weigh it down with compost.
Do your research and planning over the winter.
Plant and mulch in the spring.

But don't cheap out and try and do things the easy way.
You will be kicking yourself for years.