Select Don difficulty at the start of the game and finish the game without changing said difficulty.
This game is not particularly hard, and if you did your first run through on normal (I did not) and are doing Don as New Game+ it will be even easier as you will have a good number of abilities and will know the level layouts.
Some general tips:
I believe the three most important abilites are the ammo belt upgrade (since you will be burning ammo fast on Don and there is not much to go around on some levels), the health execution/more health from hearts upgrade, and gun channelling.
Use gun channelling in the large arena fights, as you can kill a number of the heavies very quickly, especially if you get the upgrades for duration/shooting through walls. It is also very useful for a last ditch attempt to not die when overwhelmed. It does, however, gib enemies, so I would not reccomend using it on the minions during boss fights (as you often want to kill them for health hearts).
If/when you get it, the darkling throw is very useful as he will kill _anything_ he hits, even heavily armoured enemies.
If/when you get it, swarm can be very useful on higher-level enemies. It will prevent teleporting enemes from doing so, and will cause shield enemies to drop their shield. It is also useful for crowds of basic enemies, as you can hit two or three low-level enemies, and walk in to get a few easy executions for health/ammo/etc.
If/when you get the black hole upgrade, use them as often as possible in large fights or hold onto one for use when the pressure is on. While it can clear a room on its own, it does take the bodies, so if you need hearts, be aware.
Watch out for the whip enemies, as they will pull your current weapon out of your hands. This can be very frustrating later on in the game when they steal the only weapon with ammo that you have. try to kill them from as far away as possible.
Weaponwise, I found two SMG's to be the most useful. Most enemies carry them (so non-box ammo is usually available) and you can be fairly accurate with them in short bursts. Often you will have several enemies pop up close to you and I found pistols did not offer sufficient damage to be worth the increase in accuracy. Carbine/AK/Striker is good as a backup, but I found the SMG's superior from a damage viewpoint. I did not find shotguns particularly useful even with the double-pump upgrade, but a couple of friends have said they had good results, so try them once or twice to see if they work for you.
I'll update this over the next day or so with more tips from myself or the comments.