With the many materials from which countertops can be made, there are many methods to do this. It sounds like you are replacing a laminate countertop with an integral backsplash with a similar unit and this post will advise accordingly. To remove the old one, take out the drawers and shelves. You will probably find screws at the top of the base cabinets that go thru triangular pieces in the corners and up into the old top. Remove the screws and if it doesn't come loose, keep looking since there usually is a screw or 2 you have not found. Lay the new top into position and check the fit against the wall. If it matches to within about 1/16", you are lucky and can skip the next steps. Take a "junior high school" compass/dividers and scribe the top of the backsplash with the pointer on the wall and the pencil set to the widest gap. Using a belt sander, remove material from the back of the backsplash until you achieve a perfect fit between the wall and the back of the backsplash. Replace the screws with "drywall" screws and be careful that they are not long enough to go thru the laminate. Run a small bead of caulk between the wall and the top of the backsplash and, using a wet finger, remove any excess and make it smooth. Stand back and admire your work, then make a cup of coffee, just to test it out.