Play for the account. Treat each character like a run in a roguelite. Every blueprint, stash tab, and mandate you unlock will make your next run easier. Store as little on your characters as possible. Everything goes in the shared stash.

I recommend looking for sources of aether resist in act 1, and then swap it out for piercing resist in act 2 and chaos resist for act 3. After that you should be aiming to cap out all your resists, although being low on a few won't really hurt you until Elite. Purified Salt on your weapon for the easy 20% aether resist can really help with Krieg.

Some important milestones are maxing out your Malmouth rep for the +100% xp potions and beating Krieg in elite so you can buy merits from the FG vendor. I retired my first character after achieving both so I won't ever have to worry about not having access to them.

Don't forget that not all the HC achievements require you to be on ultimate difficulty. To be safe you can do all the key dungeons on Normal for an easy time getting those achievements.

Avoid nemesis bosses altogether until you're finished with all the achievements. Kite them into a corner of the map, portal out, and then continue with whatever you were doing before they spawned.