Yakuza Kiwami

The Coliseum can be accessed in Chapter 5, when Kiryu chooses to fight in order to gain info from the Florist of Sai. It consists of tournaments consisting of 3 fights against random opponents, although the choices for each tournament remain the same. To complete all tournaments you must rank up to Rank S and finish the final tournament. I recommend starting the coliseum as soon as possible, as buying the Image items from the store will allow you to complete Komaki's training and work towards:



The points needed for each rank are as follows:
Rank E - 500
Rank D - 1400 (900 more)
Rank C - 5000 (3600 more)
Rank B - 13,000 (8000 more)
Rank A - 22,000 (9000 more)
Rank S - 42,000 (20,000 more)

To get the points you will have to participate in the following tournaments:

Exhibition Tournament - Rank F - Basic combat against rookie fighters.
Screaming GP - Rank F - Potions appear in the middle of the ring, which either fighter can pick up. Red increases attack, blue increases defense, gold makes you invincible, purple poisons you.
Scorpion GP - Rank E - Basic combat against low-level fighters.
Street Fight GP - Rank E - Weapons litter the arena in this tournament. Use Beast style for the best results.
Burning GP - Rank D - The floor gets hot in this tournament. If you or your opponent are knocked down, you will take damage.
Hyper GP - Rank D - Basic combat against intermediate opponents.
Weapon Master GP - Rank C - Fights with a preset weapon, I recommend using Knuckles as you will be locked to the Rush style. Dodge your opponent's opening attack to fill up the Heat bar, then use the + heat action for serious damage.
Breakout GP - Rank B - Same as the previous tournament but the wall is destructible and you and your opponent can fall out.
Magnum Force GP - Rank A - Basic combat against high-level opponents.
Maximum GP - Rank S - Basic combat against the best the Coliseum has to offer.

If you are struggling with the later tournaments, don't be afraid to come back later in the game once you have levelled up some more. Fully levelled Dragon style makes it a breeze. Simply switch to Rush at the start of the battle, dodge the opponent's opening attack to fill the Heat bar, then switch to Dragon style and use the Essence of Sumo Slapping twice ( + at range) to flatten your opponents. I have attached a video demonstrating this strategy.

The best way to accomplish this is by completing all the Komaki training subquests for your Dragon style, specifically to acquire the Tiger Drop maneuver, a counterattack punch that takes massive chunks of health out of opponent's life bars and can knock out your opponents in as few as three or four Tiger Drop punches.

To access to these sidequests you need to first get the Wooden Katana weapon to give to Komaki (bought from Ebisu Pawn), then after giving him the item to enable training to begin you will need to obtain all nine image items to trade with Komaki to acquire all his Dragon moves (Image of a Tiger offers you the Tiger Drop move, and is the penultimate item before the Image of a Dragon). These images are all available in the Coliseum hallway shop in Purgatory, obtained with points you earn from your performances in tournament fights. You'll get the most points from winning rounds, executing heat actions and crowd pleaser moves that pop-up and give you 10 seconds to execute, as well as bonus points for winning a round without taking any damage and a time bonus for winning a tournament within a set time limit.

When you begin your Coliseum battle, you'll start at Rank F, and you'll need to work your way up to the top rank of S in order to access all the other tournaments leading up to the Maximum GP, which also offers you the most points. Emphasize getting more and more XP for which you can build out your battle style trees, with added preference to the Rush style, as you'll gain heat for successful evasion for attacks and be able to perform the Essence of Crushing Elbow when you get enough. It's very possible to win the Maximum GP without having Tiger Drop unlocked, though it would take longer in terms of winning all three fights depending on your opponents (which I actually did believe it or not, utilizing the Essence of Sumo Slapping!).

Once you get all the points and acquire the Images to give to Komaki, complete all the training missions to acquire the special moves for the Dragon style. I would recommend first getting in some warm-up battles either in the lower ranked Coliseum tournaments or in Kamurocho against basic enemies so as to try out and perfect the Tiger Drop move. Just hold down to focus stance on your opponent, and then press the when their attack is about to land on you and execute the most devastating counterattack in the game. Try to add a bit of distance between you and your opponent so as to offer a slightly bigger time window and a better opportunity for your counter to succeed. Depending on your opponents in the Maximum GP, you can complete your run in less than a minute using Tiger Drop as well as Essence of Sumo Slapping, I was actually able to beat Komaki in just 10 seconds just from these two moves, while winning the tournament in just one minute.

EDIT: Included a video capture of my Maximum GP victory described in last paragraph.