This paper investigates a class of reaction-diffusion models that describing the chemotactic movement of cytotoxic T lymphocytes (CTLs) in HIV-1 infection, thereby incorporating nonlinear diffusion mechanisms for CD $4^{+}$T cells, infected cells, and CTLs. Assuming that the CTL diffusion coefficient satisfies $D_3(w)\ge \hat{D}(w+1{)}^{\theta }$ with $\theta >1$, and the chemotactic sensitivity function satisfies $|\chi (w)|\le \hat{\chi }w(1+w{)}^{\eta -1}$ with $\eta \le 1$, this study proves, via energy estimates, semigroup techniques, and the Moser iteration, provided $\theta >1$ and $\eta \le 1$, that the classical solutions globally-exist and are unique and uniformly bounded for any spatial dimension $n\ge 1$. The results demonstrate that a sufficiently strong nonlinear diffusion (with $\theta >1$) can effectively suppress chemotaxis-driven instability, thus ensuring the boundedness of the systems long-term behavior. This research provides mathematical theoretical support to understand the regulatory role of immune chemotactic movement in viral dynamics.