Hello from Lake County.  thank you for letting me answer your question.  It is perfectly acceptable to mulch under your willow.  And yes, you should kill the  grass or you will be fighting that forever.  that being said, you need to be careful on how you get rid of the grass.  using a product containing glyphosate could also hard the tree.  it should be sprayed only on the grass and follow the directions on the container. the label is the law.  Also if you dig the grass out, be careful not to hurt the roots.  You can smother the grass with newspaper and plastic, but again the roots need to be spared. they need to breathe.  Use an organic mulch and spread thinly--2-3 inches.  the trees roots go out as far as the canopy of the tree.  And please do not pile the mulch up on the bark.  that is called 'volcanoing'  Here are some fact sheets for you to read for more indepth info

https://extension.purdue.edu/Miami/article/12124

https://extension.psu.edu/mulching-landscape-trees

Hope this helps.  please respond if you have any other questions