According to the ingame stats menu, Melee Attack is defined as
"This is your general skill in melee combat. It governs your chance to successfully land a hit on your opponent, and is essential (to) whatever weapon you choose."
It can be increased up to 5 by using the Mk1 training dummy or 10 by using the Mk2 training dummy.
Ways to increase Melee Attack quickly
An effective yet high risk method of raising melee attack and defence quickly is by lowering your combat skills. This happens when you equip a backpack, chainmail, various helms or some armours such as the Holy Chest Plate. In combat, when your stats show a negative in brackets (e.g -10 from a traders large backpack) this is the base used to calculate your experience gain. Meaning, if you have 20 melee attack normally and find it is levelling slowly, if you equip a traders large backpack it will be counted as 10. You will then gain the amount of exp that someone with 10 melee attack would. This is also true of Melee Defence.
Pieces of equipment that lower your combat abilities are cumulative, for example a large traders backpack in addition to chainmail, the holy chest plate and a tin can helmet would lower your skills by (-18). Thus allowing for quicker exp gain.