I know someone who got that job within 4 months of graduating from the adv. ll.m at Leiden. He also had experience working for his government and had done several UN internships.
With the UN it’s usually good to have done at least one legal internship for at least 6 months. You can make this (relevant law internships) part of your degree at Leiden. Also, if you get into Leiden, make sure to apply for their IHL Clinic. Right now would also be a good time to do their MOOC too (https://www.coursera.org/learn/international-humanitarian-law).
Also, learn some languages if possible (this will really make you stand out, particularly Arabic).
Best of luck!
Ps. Berkeley also has a really good IHRL centre, might be worth applying there too.