Baldur's Gate 3

A quick way of getting this would be a martial/melee class, such as Barbarian or Monk, combined with an Elixir of Speed and Extra Attack. Pop the elixir to be able to strike twice in one turn, but doubled with Extra Attack, it'll give you four; combine that with Monk gaining Unarmed Strike as a Bonus, or any other Bonus attacks as other melee classes, and you should get this starting:

Barbarian, Fighter, Monk, Paladin, and Ranger at level 5
Bard, using the College of Valour or Swords, at level 6

Fighter also acquires Improved Extra Attack, increasing the amount of additional attacks you can do from one to two, at level 11, so don't feel like you have to solely play Monk to get this trophy.

There are a million ways to do this, but this one is super simple, doesn't require any consumable resources like potions, and for most people won't even require any specific build changes, multi-classing, re-speccing, etc. (assuming you keep Lae'zel as a fighter.)

All you need for this method is:
- A level 5 fighter of any subclass
- Two weapons that have both the light and finesse properties (daggers, shortswords, scimitars or hand crossbows)

Once you have 5 levels in fighter, you will have the only two abilities we need, which are Action Surge (from level 2) and Extra Attack (from level 5.) From there, arm the fighter with two weapons described above, and get them in a position to attack either an enemy with lots of hit points or several enemies they can reach in a turn.

Once you have all that, just do these 3 steps. I put them in my preferred order as it minimizes your chance of accidentally burning your bonus action attack sub-optimally, but they can be done in any order.

Step 1. Bonus Action Attack
I recommend this first because if you use the main hand attack first it automatically adds your Bonus Action Attack after it. By doing this one first you can choose your targets more deliberately.
Step 2. Standard Attack (twice)
This can be a standard melee or ranged attack, throw, improvised weapon, or unarmed attack. Once you do one of these the first time, they will all be highlighted in red, indicating it's letting you choose your second attack for the 'extra attack' feature.
Step 3. Action Surge & Repeat*
Hit Action Surge (a little spikey looking heart) and you will get your action back. Repeat Step 2, with both attacks.

That's 5 and you'll have your achievement.

*In normal difficulties, you could replace Action Surge with a Haste spell or potion of speed - but note if you're in Honour Mode, this will only give one extra attack, not the full 2-attack "action", which is the main reason I recommended a fighter for this.

There are any number of combinations of effects to do this, but what I'm listing below is the easiest, since it requires the fewest additional features and doesn't burn any resources like potions. All it requires is a level 5 fighter and two weapons that are both light and finesse (daggers, short swords, or hand crossbows.)

Basically you're just combining two fighter abilities: Extra Attack and Acction Surge, with dual-wielding.

So you can go....
- Bonus Action Off-Hand Attack
- Main Action attack
- Second attack
- Action Surge (get another action)
- Main action attack
- Second attack

There you go, five attacks in one turn, with no potions or anything burned, no complicated multi-classing, nothing.... just a fighter being a fighter.