Yes, kill the grass, cut a nice edge where you want the bed to be and then spray with round up. Use caution and a barrier to protect your tree. You may need to treat it for a few weeks to be sure the grass is dead, then lay down your mulch. I'm not a fan of "Weed Barrier".

'Design' guys 'design' it is in the dictionary but not in your question.

really sorry I posted in wrong forum inkognito-you don't have to be so sarcastic-I am sorry that I bothered you.

If you don't want to use Round Up you could lay down cardboard or layers of newspaper (10 pieces of newspaper thick) and put the mulch over that. It will help suffocate the grass and the cardboard/paper will decompose. Mowing the grass really short first wouldn't hurt.

If the grass is Bermuda then killing it will be a lot more difficult. Several applications of Round Up, and even then it's likely to come back.