fista1.103: Hello,
I wonder if it is possible to use two melee weapons simultaneously when playing the Warrior class?
In general, yes.
More specifically, there's the following options:
1. Pick Nightblade as second class, they have a passive that enables dual-wield melee, as well as several weapon procs that will hit with both weapons. The inquisitor class has the same for dual-wield ranged.
2. Craft a blademaster type relic (available from level 15-20 on, I think), which will also enable dual wield. There are several, with higher level requirements and better bonuses. There's also the gunslinger type, which will enable dual-wielding ranged weapons.
3. Use an item that enables dual-wield; there are a few items in game that actually allow you to dual wield, but they're pretty rare. On top of my head I can only recall a single legendary medal that does allow for DW melee, but there are several items for DW ranged.