5.un born/dead end
Check the bulletin board.
After the scene ends, talk to Rin again to get Sakura's weekly item.
Go to the Arena Entrance for an event.
Enter Arena 1F.
Move a bit forward for an event. You get Matrix Level 1.
Leave the classroom for an event.
Talk to Taiga on 1F and accept a quest to get a coin.
Talk to Sakura in the Nurse's Office to get your weekly item.
Talk to Rin(Roof)/Rani(3F).
Enter Arena 1F.
You're significantly weakened, so you should avoid fights when you can. You can make this easier by equipping Shockwave-2nd Strike and stunning any enemies that get close. Move forward until you reach an area near the center of the map for an event. Choose to run.
Rin/Rani gave explicit instructions not to look inside, so go to the Nurse's Office. Go ahead. Peek inside. You know you want to.
Go to the Library for an event with Leo.
Go to 1F for an event. You get Matrix Level 2.
If you have Caster, go to your Private Room for an important conversation. This conversation is erroneously set as a normal conversation by the game. So if you have some achievement-oriented conversations in the queue, you may have to leave and talk to her a few times before it appears. But it should be obvious when it does once you get Player Matrix Level 2.
Enter Arena 1F.
Go a little past the spot where you met Julius yesterday for an event where you have to fight him. He will only attack for his 4th action each turn. This attack is unblockable, so you need to cancel it by using an attack Skill.
Explore the rest of the dungeon. Make sure you get the Babylonian Coin just west of where you fought Julius and the 1st Trigger past the exit. Also, continue to the west of the Coin down the southwest path to find a Nephilim. Behind it is the Staff of Fortitude, one of the very few things in the game that can heal your Servant's MP.
Talk to Rin(Roof)/Rani(3F).
Give Taiga the Babylonian Coin and accept a quest to get a Lobster.
Enter Arena 2F.
Place the 3 traps in the places you're instructed to. You can't enter the northern half of the dungeon today.
There is a Nephilim at the end of the side path just east of the last trap location.
Talk to Rin(Roof)/Rani(3F).
Go to the Library and check the marked shelf for an event.
Enter Arena 2F.
Go to the locations you placed the 3 traps. After an event at the 3rd trap, you have to fight Julius. It's much more straightforward this time. You get Matrix Level 3.
You can explore the rest of the dungeon now. Don't forget to get the Spiny Lobster from the north side of the large room in the center and the 2nd Trigger east of the exit.
Go to the Library and check the marked bookshelf if you want to read more about Assassin.
Talk to Rin(Roof)/Rani(3F).
Go to 1F and talk to Rin.
Give Taiga the Spiny Lobster.
Enter Arena 2F and grind until you're satisfied.
After an event with your Servant, with Saber or Archer, you get Player Matrix Level 2. With Caster, you get Player Matrix Level 3.
Go to your Private Room and organize your information to get Matrix Level E.
You now have access to your Servant's Noble Phantasm for the first time. All three of them have restrictions on their use. For Saber, it becomes available from the 4th turn. For Archer, once your Projection Level is at 5, use Pre-Projection again to make the number start flashing. For Caster, it becomes available after using 9 Skills.
When you're finished preparing, talk to Kotomine on 1F.
I typically leave this part alone as it's totally a personal preference thing. However, in this case, for the pre-battle dialog, I would recommend choosing option 1, then 3, then 2, then 1, then either.
Assassin always buffs himself with Focused Chi for his first action. This is a prerequisite for his Noble Phantasm, so if you can prevent him from using it—by, for example, using Hack—you can stop his Noble Phantasm. If he does get a chance to use his Noble Phantasm, he always uses it for his 2nd action. It always takes 90% of your current HP if you let it hit you, so don't bother wasting a heal right before he hits you with it. It can't actually kill you no matter what. The only Focused Chi you need to stop is immediately before the Noble Phantasm. So if you can see his 2nd action and it is definitely not his Noble Phantasm, you can safely ignore Focused Chi entirely.
Other than that, there's nothing too fancy about him. He has no offensive Skills, so the only attacks he has that are capable of killing you are his basic actions.
After the battle, there will be a scene in your Private Room with your Servant. If you have Saber or Archer, you get Player Matrix Level 3. If you have Caster, there is a choice involved. Choose option 1.