The answer depends on what type of lawn grass you have. If your lawn is bermudagrass or zoysiagrass you will need to wait until it is actively growing and then spray it with a product containing glyphosate (Roundup is one such brand). Wait two weeks and then rototill the area. Then water the area and wait for resprouting after which you repeat the spray applications. Then the area will be relatively bermuda free and ready to garden.
If the grass is St. Augustine it is much easier to control. Putting a lot of topsoil on the area as you indicated will work fine. I suggest you mow it closely and then cover it with 4 or more inches of soil. I would avoid the landscape fabric under the soil you are bringing in as it won't decompose and will restrict root growth downward below the fabric.
Laying some newspaper about 6 sheets thick over the mowed turf and then covering it with some compost or mulch for about 3 weeks or so will also kill the St. Augustine. Then you can either remove the paper or just rototill it into the soil. Black plastic or landscape fabric can also be used in place of the newspaper. Spraying actively groing St. Augustine once with glyphosate will also do the trick.
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