Martyr Logarius has two distinctly different attack types: long range magic and close range melee combat. The magic attacks is mainly what Logarius uses in the first phase and has a different magic attacks that can be difficult to dodge at times. Once phase two hits, he will begin to use his scythe more for melee attacks.
Alongside his scythe, he will sometimes put down a sword that will trigger projectiles to fly at you and can mess up your attacks. The one upside is that his close range can be parried but is very tricky to do. He also has a habit of hitting through structures in the arena. He is absolutely relentless in this phase and it can be very hard to find openings to use blood vials.
The strategy that I can offer to take him down is to play a mix of both aggressive and defensive. You want to aggressively want to be close to him so he doesn't use his annoying jump attack (if he does this just dodge just before he lands) but you also want to be ready to dodge his fast combos. Pick your shots very carefully and always shoot or hit the sword as soon as Logarius puts it down because it will save you a lot of trouble. Use a fast hitting weapon as well and his crown will soon be yours.