Honestly, I'm always a bit surprised when people ask this because 'Arcane' fanfiction is literally everywhere now. The show blew up so huge that every major platform has a ton of it. Your question about focusing on character relationships is key, though—that'll steer you to the right tags. Archive of Our Own is the undisputed king for that specific, tag-driven deep dive. The tagging system is insane; you can filter for 'Jinx & Vi (Arcane)', 'Jinx/Vi (Arcane)' if you're into that ship, or even 'Jinx & Caitlyn' or 'Jinx & Silco'. People tag every dynamic imaginable. You'll find the whole spectrum, from sisterly angst to way darker explorations of their messed-up found family stuff with Silco.
I'd start with the relationship tags, then maybe add a filter for 'Character Study' or 'Slow Burn' if you want the real relationship-heavy stuff. Wattpad has a mountain of Arcane fics too, but in my experience, it's heavier on AUs—think Jinx in high school or coffee shop AUs, which can be fun but might not have the canonical depth you're after. Fanfiction.net has an older archive, so you might find some early takes from right after the show dropped, but AO3 feels like the living, breathing hub. A pro tip: on AO3, sort by kudos or bookmarks to find the community favorites; some of those long-form character explorations are heartbreakingly good. They really dig into the trauma and the broken bonds.