Minecraft wiki entry for Hardcore_mode, associated text: Close any other applications.
Weigh the advantages of features that may cause low FPS (such as particles, graphics type, etc.)
In particular, turning down the Rendering Distance (in [Esc] > Options > Video Settings) can greatly improve FPS as well as reducing lag spikes. In addition to performance, leaving the Rendering Distance at 2 effectively stops almost all mobs from spawning in low light levels, thus nullifying the greatest challenge associated with hardcore mode. Do note, however, that this may feel cheaty and may reduce the fun of the experience.
Jungles and waterfalls cause lag on low-end systems. In some circumstances, client-side chunk update lag is instantly fatal. Consider restarting the game or reducing your chunk radius if you spawn in a jungle or mountains.