When chrooting into the client, you don't have your package tree in the client chroot by default. In the installation, the server package tree (among others) is bind mounted in the client tree, which means the client chroot actually uses the server's copied package tree. The command ltsp-chroot can be used to chroot into a specified chroot.
By default it chroots into /opt/ltsp/<arch> and mounts nothing. Default behavior can be changed by using the ltsp-chroot.conf file, or by using command line arguments. The following example mounts the package dirs and chroots into /opt/ltsp-dev/amd64.
Typically mounting the package cache is distribution specific. This function typically bind mounts the server directories containing the available packages. The mount_package_cache() function can be defined in the distro specific extension for ltsp-chroot.