UnderMine

You can do this at any time after already having completed the game (killed all named bosses in the achievement description). Technically you could do it on your very first run through the game but it would be extremely difficult and I wouldn't recommend it. Come back after some upgrades.

Best way to do this is to buy as many permanent upgrades as possible before attempting.

Also be aware of the Summoning Stone (difficulty adjustment) dial found in the pause menu after completing the game once already (only adjustable while in the hub world)! If you have already completed the game once or more to farm gold for upgrades, be sure to adjust the "Summoning Stone" (difficulty) dial in the pause menu back down to zero. This will give you the easiest versions of all the bosses in your next run.

Some other tips: If you have not yet crafted the Hungry Ghost relic, do so before your first attempt at this achievement so you can start the run with it. Its very helpful on long runs and some boss minions will heal you when killed. Enabling you to heal (a little) during some boss fights.

If you can find the Wet Blanket during your run, it will help immensely when fighting the final boss, Seer (the fiery salamander). It will save you from getting set on fire a certain number of times.

Pocket Grill also made this long run much easier enabling food to heal more.

Totem of Life would be another helpful one for a long run.

Also, if you have unlocked any alternate familiars, consider changing out the canary for one of the others. I had Firebird so I picked that. You want to use a damage or healing focused familiar. The canary is virtually useless for this run because it focuses on making money, and we don't need that on this run.

I had most of my equipment upgraded to the 2nd highest upgrade level before trying this. It was still challenging but I was able to do it on my first attempt, with luck and RNG on my side!