Upgrade to Zane, the second character, as quickly as possible. Default hero Biff has got a useless jumping arc. This makes pits and spikes incredibly hard to dodge. If you have an automatic weapon, just hold down the 'fire' button at all times. The sound is annoying as hell, but at least you'll have the upper hand on zombies. Your slide move only works for a short distance, so stand up and start sliding again if there's a gap between hanging spikes.