I haven't reached the second boss yet, still exploring.
You don't have to redo the first boss again, you can run straight too the portal although I think it's a good idea to pick up a few health upgrades first. There's a proficiency pickup at the start of 2 that takes you up to level 5 I think.
Edit: also stuff you unlock in the second biome (guns etc.) can now drop in the first biome, which can make things easier.