Back in season one of 13 Reasons Why, Clay's parents ask him if he wants to go back on his medication after Hannah Baker's suicide. They're concerned that his nightmares may be coming back, and offer him the option of speaking to someone who isn't a parent, in this case, a therapist. Though the issue of Clay's mental health comes back into play throughout the series, we're never given a full explanation.
Spoilers for season two below!
Clay's girlfriend Skye is specifically diagnosed with bipolar disorder after her hospitalisation, but we don't ever get an explicit diagnosis for Clay. It's possible from his anti-social tendencies — even before Hannah's death — that he has generalised anxiety disorder, but his demeanour doesn't seem out of the realm of normal teenage behaviour. His parents may have felt a need to ameliorate the sharper edges of anxiety to make his life easier (not to say that isn't a valid reason for medication). At one point, his mother even puts a pill bottle of Duloxetine next to him at breakfast, a drug for anxiety and/or depression.
Clay is adamant about getting justice for Hannah, even going so far as to taking a beating from rapist Bryce in order to get a confession, then eventually taking a loaded gun to Bryce's house. It's an extreme measure, and Clay clearly isn't of sound mind.
In season two, Clay starts having visions of Hannah, interrupting his attempts at intimacy with Skye. It could be seen as distressing psychosis or obsession, but extreme grief can also distort one's mindset. Hallucinations of dead loved ones is a common phenomenon, even for those without a mental illness, and also with those experiencing PTSD. It can be a way the mind deals with an unexpected and overwhelming loss. Whatever the issue, it's more than common anxiety.
Clay is just one of the many complex characters on 13 Reasons Why who are more complicated and layered than they seem to be at first sight. Perhaps we will get a firm answer on what exactly is going on with him in the potential third season.