What is the max amount of achievements I can get by performing any one action?

The Soldier can put together a pretty good chunk - though as you might expect, it's rather difficult to set up:

• Be on a server where Halloween mode is on for any reason.
• The map is Dustbowl or Gravel Pit.
• Have a crowd of enemies:
  1. A Pyro, who:
     • Reflected one of your rockets recently, and
     • has set you on fire
  2. A Demoman,
     • Wearing a Gibus
     • and Pyrovision Goggles
     • whom you're one kill away from dominating
  3. A Sniper
     • with a Balloonicorn
     • who's dominating you
  4. A Spy, who:
     • just stabbed one of your teammates
     • is disguised as you
  5. A fifth enemy, with 270 health or less.
• These enemies are all capping the first point during overtime.
• These enemies have all recently damaged the Medic healing you.

Then:

1. The Medic pops Kritz on you
2. You fire a crit rocket
3. The Sniper kills you
4. Your crocket kills all of your enemies, which wins your team the game.

As a result of this action you claim:

• Your 1000th Soldier kill
• 500th Demoman kill
• 79th Balloonicorn-owner kill
• 30th point defense
• 20th kill while on fire
• 20th game win
• 10th top-3 score for MVP
• 5th revenge kill
• 1st domination on people wearing:
  1. A Gibus
  2. Pyrovison Goggles
• 1st time ever completing a full round on either one of Dustbowl/Gravel Pit (having now played full rounds on all of 2Fort, Dustbowl, Granary, Gravel Pit, Hydro, and cp_Well).

By doing this, you get the following achievements:

• A Fresh Pair of Eyes
• Backdraft Dodger
• Costume Contest
• Dynasty
• Frags of our Fathers
• Ghastly Gibus Grab
• Hamburger Hill
• Impenetrable Defense or Impossible Defense (depends on map)
• Medals of Honor
• Mutually Assured Destruction
• Nemesis
• Out, Damned Scot!
• Semper Fry
• *S*M*A*S*H*
• The Great Deflate
• Tri-Splatteral Damage
• War Crime and Punishment
• War Crime Spybunal
• World Traveler
• One (if not two!) of the Soldier milestone achivements

This gives you a total of 19+2.

I think I've found another one:

You are a Spy disguised as an enemy Spy backstabbing that same Spy who is

• On your friends list
• Has his revolver out
• Is dominating 3 of your teammates
• Is also disguised as a Spy
• Is equipping the Balloonicorn and is your 79th Balloonicorn victim

You get Revenge on that Spy and get your 1000th stab on top of a Control Point that just became available because the other spy captured it while being healed by a medic and you being covered in Jarate and after having that enemy spy trigger your Dead Ringer and you exiting a teleporter and having 4 prior spy achievements, and this kill is your 5th kill in a row without dying, gives you:

• Come in from the Cold (revenge)
• High Value Target (stab someone with 3 dominations)
• Spymaster (1000 stabs)
• Agent Provacateur (stab 10 friends)
• Identity Theft (stab the guy you are disguised as)
• A Cut Above (kill a revolver-wielding spy with your knife)
• Insurance Fraud (get a backstab while an enemy medic heals you)
• Sleeps Agent (kill the guy that triggered your DR within 20 seconds)
• On Her Majesty's Secret Surface (start capping a point as soon as it becomes available)
• Counter Espionage (backstab a disguised spy)
• Wetwork (kill someone while jarated)
• Point Breaker
• Milestone 1
• Milestone 2
• Milestone 3
• Costume Contest (kill a spy disguised as your current class)
• Nemesis (5 revenge kills)
• Riftwalker (get a kill shortly after exiting a tele)
• Hard to kill (5 kills in a row without dying)
• Hardcore (get 1000 total kills)
• The great deflate (kill 79 people with balloonicorns)

Total achievements from 1 stab: 21 achievements

Apparently 25 is possible as Sniper, courtesy of Reddit:

Setup

You must have exactly:

• 4 Sniper achievements
• 999 kills (all Sniper)
• 1 prior Domination
• 78 Balloonicorn kills
• 24 parachute headshots
• 9 Sniper headshots
• 4 prior Revenge kills

Equipment

• Pyrovision goggles
• The Machina
• The Razorback

Action time!

1. Take a teleporter
2. Take a fully-charged shot
3. Your fully charged shot must kill all of the following:
   • A sniper,
     • Whom you've killed 3 times
     • Who is wearing
       • Pyrovision
       • A Balloonicorn
   • A Spy,
     • Who recently tried to backstab you
     • who is now cloaked
   • A second Spy who just uncloaked
   • A Demoman
     • Who is parachuting with the Base Jumper, following a sticky jump
     • With a headshot
   • A second Demoman
     • Who is charging towards you
     • Who is also dominating you
   • A Medic
     • With full Uber
   • A Scout
     • During a double-jump
   • A sentry gun
4. Taunt (well, who wouldn't after all that?)

Sniper-specific Achievements

• Australian Rules
• Be Efficient
• De-sentry-lized
• Dropped Dead
• Flight Cancelled
• Jumper Stumper
• Kill Everyone You Meet
• Not a Crazed Gunman, Dad
• Parting Shot
• Self-destruct Sequence
• Shock Treatment
• Shoot the Breeze
• Socket to Him
• The Last Wave
• Triple Prey
• Überectomy
• Sniper Milestone 1

General achievements

• Nemesis
• Riftwalker
• The Great Deflate
• Hardcore
• Hard to Kill
• Emergency Brake
• BFF²
• A Fresh Pair of Eyes

Total: 25 achievements