A file system can be thought of having three (3) types of data: file data, file meta-data and file system meta-data. File data is file or directory contents. File meta-data is that which tells us where the file data is stored on the disk. File system meta-data tells us how the file system allocates the blocks used in the file system.
The FAT file system however does not keep the lines so clear cut. Its disk structures often blur these distinctions.
The File Allocation Table (FAT) itself blurs the lines of the file meta-data and file system meta-data. That is, the FAT entries identify both the cluster number of the where the next cluster of file (or directory) data can be found as well as indicating to the file system whether the cluster identified by the index into the FAT is available (or not). As you indicated in your question, this forms a chain. A special marker (the specific value escapes my memory) indicates that the cluster identified by the index into the FAT is the last cluster in the chain.
Directory entries in a FAT based file system are both file data and file meta-data. They read like files with their entries being the "file data". However, their entries are also interpreted as file meta-data, for they contain the file attributes (permissions, file size, and the starting cluster number--which is an index into the FAT).
The root directory is a special directory on a FAT file system. If memory serves, it does not have either a "." nor a ".." entry. On FAT12 and FAT16 systems, the size of the root directory is specified when the disk is formatted and is thus of fixed size--however, its clusters are still marked in the FAT. On FAT32, the root directory size is not set at format time and can grow. The starting cluster of the root directory is stored in a special field in one of the file system meta-data structures (as I'm going by memory the name of this structure eludes me).
Hope this helps.