The exact method for replacing the parts of a Delta faucet depends on the model, but it typically involves locating the needed parts, disassembling the faucet and replacing the affected part. The first step is identifying the area that's leaking.
If the faucet is leaking from under the handle, the stem unit assembly needs to be replaced. DeltaFaucet.com offers a locator for finding this part, both in stores and online. Once you have the parts, turn off the water shutoff valve near the faucet. Drain the remaining water in the lines.
Remove the decorative caps using a flat-head screwdriver. Unscrew the unit assembly. Pad pliers with electrical tape to remove the handle from the mount. Unscrew the packing nut followed by the stem assembly. Replace the packing nut with the new one. Clean out the interior of the handle, and put it all back together.
If the faucet is leaking from the aerator, the seats and springs need to be replaced. After replacing the packing nut, locate the springs inside the assembly. Replace these with the new ones in your kit before putting the assembly back together. Pull out the washer seat from the front of the faucet, and replace it with the one in your kit.