Do a search here on removing bermuda grass to get all the pros and cons. This has been discussed quite frequently here in the past, maybe also go the the forum on lawns. In short, even with Roundup being used, it isn't a one shot deal when treating bermuda grass, and just tilling the soil will only encourage it and make it that much more difficult to remove. Any broken piece of rhizome will regrow. Roundup has no residual effect on soil fertility, but people will debate about its chemical persistence in the soil.

Personally, I wouldn't even attempt to get rid of bermuda grass without using herbicides. It will also probable take several repeated applications alternating between watering to encourage new growth, spraying, waiting for die off and re-growth, and repeating the process once or twice more until you are certain it is all dead. Anything less thorough is pretty much a guarantee that it will be right back.

You'll also need to consider how to contain the lawn edge to keep it out of the new planting area; usually a foot deep by 8 inch wide concrete mowband is the most successful method for establishing a barrier deep enough to prevent underground recolonization by rhizomes, and will allow you to use a weed trimmer to keep the above ground rhizomes out.

I'd also suggest that you keep any new plantings at least a foot clear of all lawn edges, so that you can more easily see bermuda grass shoots if/when it starts to creep back in. You'll have to keep ever vigilant if you don't want this new bed infested. Maybe not as much an issue if you are only planting annuals that will be replaced seasonally, but if using perennials and shrubs, it is a royal P.I.T.A. to weed out bermuda grass once it has gotten established amongst permanent plants.

I agree with bahia except for one thing. The mowband might be the most effective deterent for regrowth, but it's both expensive and limiting. I've expanded the depth of all my beds several times. With a mowband, you're stuck.

Bermuda is a PITA. I use a trench edger and dig out all pieces of bermuda as they encroach, or spray with glysophate when no other plants are close enough to be effected.

Now that the bermuda is actively growing, follow bahia's guidelines.

HTH, Rosie