assassin is allready a sub-class.
fighter is the general description for champions who want to be in or near melee range for an extended period of time. for example: garen. he is a prime example of a fighter.
now dividing fighers into sub-classes like juggernauts etc is generally okay but not really useful due to the lack of distinct roles of most champions (a good thing).
for example, pantheon is a fighter because he wants to be in near-melee range for extended periods but he is also an assassin because his damage is one quick combo and he has no abilities that support building defensive items which "forces" him to build damage and be squishy so in a duel, he is rather weak as all assassins are and his only chance to deal his damage properly is to catch someone off guard, i.e. assassinate.
leblanc is a mage as she uses spells and ability power aswell as mana but her abilities are one quick burst of damage on a single target/very small aoe so she is mainly used to kill a single target very fast.
viktor is a mage too for the same reasons and his combo can deal just as much damage, if not more, just as fast as leblanc and has alot more aoe, utility and range. what he doesnt have tho is the mobility and thus, he is not an assassin because the area around him in which people count as "in kill range" is much lower than any assassins. he also has no way to get in and out quickly, so he cant be an assassin.
zed and fizz are mainly assassins as both of them deal their damage quickly to one target but dont neccessarily want to stay after using their cooldowns.
there sure are different types of assassins but they all generally do the same: put themselves up close and kill quickly, then get out.
no real need for sub-classes. all assassins are somewhat mobility assassins. malzahar is not an assassin, he is a mage and even tho he can assassinate very good, he simply isnt one. if you facecheck a garen, you will pretty much die and he can assassinate wonderfully but he isnt an assassin either.