Yu-Gi-Oh! Legacy of the Duelist

To unlock all of the challenges you need to first complete every duel in the each single player campaign and the reverse duels as well.

After that the quickest way to beat each challenge deck, that I've found so far, is an Exodia deck. I modeled my Exodia deck after the Exodia cheat deck you get when you beat Seeker. I changed a couple cards around since I didn't own a few, but it still revolves around the concept of milling through your deck as fast as possible. With the deck I had I was able to routinely win matches in 3-4 turns with practically no effort and it's possible to win before your opponent even has a turn. Choosing to go first after the rock-paper-scissors is also helpful to getting those 1 turn wins. I was able to get through 10 or so challenge duels in about 20 minutes as opposed to a 3-5 duels with my Fire Fist deck. And the best thing about this deck, minimal attention and skill is required. Honestly, I didn't need to pay attention to nearly any of my opponents plays; just focused entirely on my cards.

Here's my deck list:

Exodia the Forbidden One x3
Right Leg of the Forbidden One x3
Left Leg of the Forbidden One x3
Right Arm of the Forbidden One x3
Left Arm of the Forbidden One x3
Emissary of the Afterlife x2
Sangan x1

Pot of Greed x3
Swords of Revealing Light x3
Pot of Duality x3
One Day of Peace x3
Graceful Charity x3
Dark World Dealings x2
Heart of the Underdog x2 (I'd remove these if you have more Upstart Goblin cards, only owned one)
Upstart Goblin x1
Dark Hole x1
Raigeki x1

Gagaga Cowboy x2 (not really needed, but can help in a pinch)

I'll be modifying it too as I get more and better cards.

Here's my deck to easily win every duel near enough first turn, it's a fully optimised Exodia deck which rarely loses due to bad hands. You can easily draw all exodia card first turn and it requires no skill.

3x Pot Of Greed - Obtained from Seeker
2x Graceful charity - Obtained from Seeker
3x Reckless Greed - Obtained from Seeker
2x Dark world dealings - Bastion Booster Pack
3x Broww, huntsman of dark world - Bastion Booster pack (Rare)
3x Upstart goblin - Obtained from Seeker
3x Allure of darkness - Obtained from Yubel/Jesse Anderson Booster pack
3x Toon table of Contents - Seto Kaiba Booster pack/Obtained from Maximillion Pegasus
1x Blue eyes toon dragon - Seto Kaiba booster pack (Rare)
3x Destiny hero - Dogma - Jaden Yuki Booster pack (Rare)
3x Destiny hero - Plasma - Jaden Yuki Booster pack (Rare)
3x Destiny draw - Jaden Yuki Booster pack
3x Trade-in - Seto Kaiba booster pack (Rare)
1x each of the Exodia pieces (left leg, right leg, left hand, right hand, head) - Obtained from Seeker

Total cards: 40.

The deck is an "Auto-pilot" deck, very little skill is needed to play the deck, although there are some tips which will help you to win even faster, here I'll explain some tactics:

Whenever you draw Toon Table of Contents, always use it to obtain your second Toon Table of Contents, then use that to obtain your third Toon Table of Contents and then finally use the final Toon Table of Contents to add Blue-Eyes Toon dragon to your hand.

Whenever you use "Dark World Dealing" or "Graceful Charity" always discard "Broww, Huntsman of Dark world" in your hand, this will allow you to draw an extra card (This is due to Broww, Huntsman of Dark world's effect, hence why he is in the deck).

Never summon a monster, it's not worth it in my opinion.

Finally, when it comes to Reckless Greed, once you set them down, it's best to wait until your Draw phase, once you draw, then activate your set Reckless Greeds, that way you have one extra card in your hand to finish off the duel. If your opponent attempts to activate Heavy storm, or any other card to destroy your set Reckless Greed trap cards, chain them in response, this way you get the effect of Reckless Greed even if they get destroyed (Unless your opponent responds with "Trap Jammer", or any other card which negates Reckless Greed's activation).

Finally, I'll do a more detailed breakdown on what I mentioned above to give new players an idea of what spell card can be used with each monster card:

Destiny Draw (Spell card) - Discard 1 Destiny Hero card and Draw 2 cards - Discard Either Destiny HERO - Dogma or Destiny Hero - Plasma, since you have 3 copies of each, there are six (6) targets for Destiny draw.

Allure of Darkness (spell card) - Draw 2 cards, then banish 1 Dark monster in your hand - Banish Destiny HERO - Dogma, Destiny Hero - Plasma or Broww, Huntsman of dark world. Since you run 3 copies of each card, you have nine (9) targets for Allure of Darkness (technically you have 14, since the Exodia pieces are Dark attribute cards, but you'll never banish them).

Trade-In (spell card) - Discard a level 8 Monster from your hand, Draw 2 cards - Destiny HERO - Dogma, Destiny Hero - Plasma and Blue eyes toon Dragon are cards you can use with this, you have 3 copies of each Destiny HERO and 1 Blue eyes, thus you have seven (7) Targets for Trade-in

Toon Table of Contents (spell card) - I know this can get monotonous fast, but the whole idea of the Toon Table loop is to mill cards from your deck, furthermore, at the end of it you get a blue eyes toon dragon which can be used with Trade-in or discarded with Dark world dealing/Graceful charity

Graceful charity/Dark world dealings (spell card) - Ideally you want to use these cards to discard "Dead" cards in your hand, such as if you have 2 Toon table of contents in your hand, you can then discard 1 of them. Ideally, you want to discard Broww, his effect allows you to draw an extra card.

Overall, I believe this deck is better than the one currently listed in the top solution, which is prone to constant "Dead drawing", so if you want you can use this instead.