Clay has plenty of motive to take out Bryce from the get-go, given his history with Hannah and the fact he's the one who records Bryce's confession back in season one (and subsequently gets beaten to a pulp). But that being said, season two ends with Clay seemingly in a good place and on the road to healing and recovery after Hannah's memorial. It seemed highly unlikely that he'd snap and murder Bryce.
Season three picks up eight months after the devastating events of season two, when Clay prevented Tyler from committing a mass shooting at the Spring Fling, and then toggles back and forth in time. Clay struggles to shoulder the burden of the cover-up while helping Tyler recover, and isn't in the best place mentally. His behaviour and abrasive past with Bryce leads the police to think that Clay murdered Bryce, and Mrs. Walker even heard him say that he wanted to kill her son the night of the Homecoming game (it was due to Bryce taunting him about Ani, though).
The police end up arresting him after finding a text in which Clay also told Bryce he wanted to kill him (Really, Clay?), but in the end, as Ani tells Sheriff Standall, Clay is innocent.