As far as I am aware, there are no items in the vanilla game (or official DLC) that provide a constant "on fire" visual or mechanical effect (the Ebony Mail artifact armor damages nearby creatures during combat / while crouching, but it's poison-based, not fire-based).
Items enchanted with a fire damage effect glow red, but the effect is subtle and certainly doesn't give the appearance of a "flaming sword" or the like. Armor, further, cannot be enchanted thusly (although other enchantments, such as the destruction bonus, do cause a similar subtle aura).
The closest I think you can get to this without mods is to use the enchanting/alchemy loop to craft yourself some gear with 100% destruction cost reduction and cast the Flame Cloak spell as-needed.
If mods are an option for you (you don't really mention your platform) you may try searching the Steam Workshop or the Skyrim Nexus. For example, I found this one, although I've never actually used it and it's not immediately clear it would provide what you want.