I find that my Weber grills (18in Kettle, Summit 450 and Summit 650), which I do not cover get pretty dirty over time. A Twitter post by @saucysue2010 on a way to recommend cleaning your Weber, they suggest warm soapy water for cleaning the outside of the grill. I have found that Simple Green works really well as it cuts the grease and leaves the stainless steel and plastic squeaky clean!
As far as cleaning off the grates before your next cook, they recommend putting your barbecue on full blast (high) or leave the vents open on your charcoal grill for 15 mins with no food on it and the lid down. The barbecue will get way too hot for cooking anything but this does burn off all food left on the grate from last cook. Then all you do is get a T-Brush or Grill Friends Grill Brush (Brush with very heavy duty wire bristles) and brush off anything on the grate and you good to go. Make sure you turn down the barbecue before putting your food on.
Editors Note: If you have some really serious food leftovers, try laying some sheets of aluminum foil, shiny side facing the burners and grates and close the lid. This will direct the heat back down and your grill should burn off things a lot quicker and more efficiently. Keep an eye on things as this will really get things burning!