From the start you will have a wrench and the Electric Bolt Plasmid in your inventory. Using these in rapid succession against most Splicers can often result in an fast kill. First, shock the Splicer. Before the convulsion ends, switch out the wrench, aim high, and hit the Splicer over the head. However, best way to eliminate the agile Spider Splicers is to unleash a few clips of anti-personnel ammo, preferably from the machine gun.