Melee Attack is levelled by defeating Monsters. Combat happens every minute and your total damage is calculated as follows:
Total attack value = Melee Attack Level + Equipment bonuses
Alpha Combat = 1 (constant value that may change in the future)
Beta Combat = 1 (constant value that may change in the future)
Damage per minute = (Total attack value * Alpha Combat) + (((Total attack value * 2) - Monster Melee Defence) * Beta Combat)
If your Damage per minute is less than 1, then it will be rounded up to 1.
Example, as a character with level 1 stats and no equipment except the default Bronze Sword fighting the Natuow, you end up with:
Total attack value = 1 + 1
Damage per minute = (2 * 1) + (((1 * 2) - 0) * 1) = 4
Doing 4 Damage per minute to a Natuow, which has 20 Health, means you will defeat 1 Natuow every 5 minutes, or 12 per hour. Remember that Monsters also attack back every minute too so you will need Food to stay alive!