Just wanted to add some advice for anyone struggling with a few of these missions, as some had me wanting to go insane.
Brawl challenge 10 - For this you need to kill 5 enemies as Kiryu without getting hit once. I found a helpful way for this was to immediately grappled the enemy in front, and throw him left. You should then be able to knock out the two weak guys to the right before getting attacked.
Ideally then focus all your attention to the guy with the pony tail, as he can often attack out of nowhere whilst you are beating on anyone else. With him out of the way, the other 2 should be pretty easy to deal with.
Trial challenge 6 - This one requires you to prevent Minami from picking up either microphone as Akiyama. Now I know the general trick is to stand between him and the microphone, thus altering his course towards the other. However I found many a time he would bypass me and still pick it up.
The thing that worked for me was to lure him as close as you can to one microphone, then immediately pick it up and throw it in the opposite corner to him. Then move to the next one and ideally throw it opposite where you threw the first one. If you can get lucky with your throw the microphones will be right in each corner.
This should now give you a little more time to position yourself, as it gives you more space to work with than the default positions they start with. Now you can hopefully run between him and each microphone as per usual, (don't even stand still), and eventually you should do it. Not saying it will work for everyone, but worth a try as it helped me at least.
Trial challenge 10 - Here you have to finish off 4 enemies by only using the arrest move as Tanimura. Now I spent a little while trying to find a good way for this, and the following worked for me well.
If you build your heat gauge up to near enough max, you can use the heat move on an enemy when grappling him from behind, (forget the name sorry). This should do a reasonable amount of damage, but more importantly, will stun him. Assuming you have enough heat left, pressing triangle should arrest that guy.
You can then rinse and repeat, or failing that, try to do small damaging moves like the dropkick or a simple throw to knock them down. If you see an enemy on the ground writhing around, that is your time to arrest them with triangle. Do be careful doing too many throws or extended heat moves as that tends to kill an enemy too fast.
Chase challenge 4 - Now I DESPISED this challenge, until I found a way that made it soooo much easier. You have to repeat the boat chase against Sugiuchi, but this time with a bunch of obstacles, a decreasing bar, and being made to twist and turn every 5 seconds.
Now at first I was shooting every moment, trying to ram him in the back, (easy to make a joke from that I know), and staying right next to him, but each time I would crash into something, or just have not enough time.
Here's the tip, all you need to do is ram into the side of his boat at about a 45 degree angle. If you get this right it should take about 3-4 rams to finish him off, no shooting, nothing else. I made a right mess of this, and wasted about half the stage, to then try this and amazingly it worked that attempt. So don't give up if you struggle at the start as you actually have more time than you imagine.
Ultimate challenge 5 - For the big finale you need to repeat all of the Ultimate challenges you just did, 1-4, all in one go. I actually did this pretty easily thanks to spending so much time already working on them, but for people having issues, here's the best advice I can give.
For the Akiyama fights, you really can just constantly spam his kicks, ideally in a corner. I was amazed how quickly someone like Minami goes down by just pressing square so much. You can throw in some triangle kicks to end the combo, and of course use up any heat actions you get during those times. But this really shouldn't be too bad for you.
For Saejima, rather like Akiyama, you really only need to spam his rush combo using square then holding triangle for the three attacks afterward. This should make rather quick work of both Sajiou the Guard, and Majima, especially if you are in heat, meaning you shouldn't really get knocked down. The Kiryu fight could cause a little trouble, but I found it best to use Guard parry a lot (L1+triangle), or throw moves.
Tanimura.... oh where do I start. This gave me the most trouble, simply because I was generally awful on the first fight against the group of enemies. I would recommend not trying to rush this first fight, as you can very easily get surrounded and wiped out fast if you are not careful. Best advice here is to target the gun enemy, and keep pounding on him whilst no-one else is around, particularly the sword enemy. Throwing the weaker enemies around, either into each other, or the two weapon wielders, can really help, as of course can good usage of his parry, which works well here.
The other two fights against a few enemies holding the briefcase, and Sugiuchi are actually rather easy, so long as you again try to be a little more aggressive trying to save up time for the Kiryu section next.
Now the last few fights involve some fights against enemies in the coliseum. The last two I think are pretty simple, especially using a lot of throws to take them down, or heat grapples against the steel walls or floor. The last guy does dodge quite often, so try to grab him after he performs a combo, that way making sure you don't miss and fall victim to his attacks.
Now the first fight against Ivar is more tricky as he is really built like a powerful tank, and does some reasonable damage too. Best advice here is to actually knock away his health with a few punches or kicks here and there, rather than doing a full square/triangle combo. I found doing this often resulted in getting smacked on my face too often, and wasting precious time. Oh and using the Komaki Dharma roll, which is doing a forward roll and then pressing trangle in heat mode. This can be used more or less anytime, as can really be helpful in chipping away.
Now to mix things up and make you worry more, you will then need to face off against Komaki himself as Kiryu, just when you figured you were done. I actually did this first time with around 2 minutes remaining at this stage. Don't worry he isn't as hard as he used to be, but can still throw you around with ease. I found being rather offensive worked well, as did using his own Komaki parry on him. This would push him back, and set him up for a few hits, to then rinse and repeat. Just try not to run around like a headless chicken if you are low on time, as hard as that may be, stay focused, and fight him like you have most other enemies thus far.
And that should be your trophy hopefully. Also remember that you do regain health during this last challenge, so again try to be a little more aggressive than you might, as I think time is the real issue here than dying. Good luck all!