The trickiest part of caulking any part of a gutter is making sure the surface remains clean and dry during the application process; otherwise, the caulk doesn't properly set and the leak may not be plugged. Wait for a string of sunny days to caulk the top of your gutter, with little to no chance of rain. This gives you time to clean the gutter, let it dry, then apply the caulk, which can take up to 48 hours to dry depending on what type you're using.
Once the gutter is clean and dry, roughen the area around the leak with sandpaper or a wire brush. This grooves the surface so that once applied, the caulk clings better to the gutter and adheres more tightly around the leak.
Apply plenty of caulk to the leak, using a caulk gun or brushing it on if it is a kind that can be painted. Cover both sides of the leak with duct tape, trapping the caulk inside so it can't flow out before it dries. Let the caulk dry for at least 48 hours before removing the tape.
Most hardware stores carry multiple types of gutter caulk, sometimes called sealant. Rubber-based caulks form a tough yet flexible seal that expands or contracts with changing temperatures without reopening. Polyurethane caulk is a good choice for gutters, although it is slightly more expensive than rubber-based caulk. Silicone caulk creates a strong, watertight seal but can't be painted on in its pure form. Any of these caulks work well if properly applied under clean and dry conditions.