Ultimately, alignment isn't necessarily how the PC feels about Chaos, it's about how Chaos feels about the PC. If you're aligned, then one of the Chaos Gods has decided to LIKE you, and they're not particularly interested in your opinion on the matter. Similarly you may think Slaanesh is the Bee's Knees, but if you're unaligned, Slaanesh doesn't think anything much about you.
That said, there's no reason fully embracing Chaos or one of the gods needs to be over-the-top cartoonish. You can do it in very deep and serious ways by delving instead into the consciousness, thoughts, and feelings that come with that kind of devotion and how the PC, as a person, experiences that. It can be a complex and nuanced experience, it doesn't have to just be "BLOOD FOR THE BLOOD GOD!" all the time.