We present new Keck/HIRES observations of seven of the highest-redshift known quasars. These include four with $z_{\rm qso}\ge 5.8$ and two with $z_{\rm qso}\ge 6.3$. The data will be used to produced a complete statistical description of the Ly$\alpha$ forest at $z > 5$, allowing us to better assess the significance of the strong absorption seen at $z>6.2$. We introduce the statistics of transmission gaps as a means of characterising the high-redshift evolution of the forest. In addition, we identify several clear absorption lines of low optical depth in the quasar proximity regions and briefly discuss using these to place constraints on the thermal history of the IGM at $z>5$.