The uniques are not needed for lvling, but if you have them you should use them as it will make lvling a lot faster.
If you're lvling from a fresh start, then constantly check the vendor for a 2 socket bow and put any 2 elemental damage runes in it (lightning prefer), this will heavily boost your damage early on. It's very important to keep your bow upgraded throughout the campaign as it's your main source of damage.
Quiver is also extremely important for damage, specially if you're using
Widowhail to level, as it scales based on your quiver. You want damage to attacks and increased bow damage on your quiver. The unique quiver
Asphyxia's Wrath can be swapped out rather early, but it's good early on.
Blackheart is pretty much only good for act 1, swap them out in act 2 or 3. you'll want Resist/flat ele damage on rare rings.
For other gear, look for the basics like Life, Resistances, and Evasion. On boots make sure to always have movement speed. Put resist in your sockets on gear. You can also use
Goldrim to help with resist.
Ideally you don't replace
Painter's Servant,
Foxshade,
Wanderlust and
Thrillsteel for the whole campaign. You want to go as fast as possible.
Surefooted Sigil can be replaced if you don't like the playstyle of constantly dodge rolling.
For amulet anoint
Acceleration or
Step Like Mist. You will eventually take acceleration, so step like mist is the better option if you're not swapping amulets later on.