Goliaths are the large giants with a weird tentacle looking hand. They will perform a ground pound that can do a lot of damage. Killing them with a melee isn't hard, but you will need to wear them down to the point where the goliath is almost dead and then deliver the killing blow. It can be annoying to get this one right as you might not leave enough health and kill them with a bullet by accident. Doing this on the easiest difficulty is of course advised, this will let your melee attacks do additional damage.
In solo play, finish off a Goliath with a melee attack