The only way to upgrade your trusty Leviathan Axe is through the acquisition and use of the crafting resource known as Frozen Flames. You'll get one almost immediately, after meeting Brok the Blacksmith for the first time, but after that, you'll need to defeat bosses, and each boss is good for one Frozen Flame. Regardless of side mission completion, you can upgrade your axe to level 5 by completing the story missions. If you want to get your axe to its face chopping best (lvl 6), you'll have to head to a shop and look under the Buy-Resources tabs, but you have to have the required materials to get it.
To purchase the final frozen flame, you'll have to meet a couple intertwined requirements. You'll need the Chilling Mist Of Niflheim to trade for the frozen flame, but the only way to get the chilling mist is to pop open the chest that sits in the central chamber in the Niflheim realm, but you have to pony up 5,000 Mist Echoes to open it, and you can't get into the central chamber until you complete the first favor that Sindri asks of you. Once the chamber is open, you can cruise the labyrinth area in this area to grind for mist echoes until you get five thousand of them. Then, pop the chest to get the Chilling Mist Of Niflheim, head back to the shop and exchange the chilling mist for the frozen flame.