Yes to becoming a Lich, no to your type of Lich. Here are my ideas.
Necromancer's GuildNot only should there be a guild of Necromancy, but you must beat their quest chain to even begin the massive quest to become a Lich. In this quest you should be forced to find the relics of Mannimarco (Bloodworm Helm, Necromancer's Amulet and Staff of Worms), create a phylactery and perform some ritual. It should be hard as hell. You should at very least have to have a reasonable level in Alteration, Enchanting and Conjuration.
Lichdom: On PhylacteriesPhylacteries are the source of a Lich's "immortality." They don't make Liches unkillable, but as long as they exist death is a far more temporary. For the sake of NPC Phylacteries this revival takes only a few minutes, but most phylacteries are kept in a Chamber (Liches resurrect at their Phylacteries) in each dungeon. Not all NPC Liches have phylacteries, those that do should look different.
Your PhylacteryYou have two options with your Phylactery, keep it on you or leave it somewhere secure. The Phylactery is the biggest upside to being a Lich, but its powers are still 'attuning' to you and you have set it up so it doesn't take too much of a toll on your body, so it takes a lot longer than most phylacteries and has a cooldown. The cooldown is at least a week (perhaps more). If you keep your phylactery on you, it will revive you at part health, blow all your greater power cooldowns and leave you silenced for 10 seconds. If you leave it somewhere secure you will be revived a week later with only very basic gear and have to retrieve your gear off of your corpse (an activator, not a container so you cannot be looted by anyone else). Though it also burns your cooldown there is a spell that allows you to teleport (gear and all) to your Phylactery.
However if you revive in Combat or someone sees you loot your old body there is a chance people will begin hunting your Phylactery. If you keep it on you they will come after you, if you leave it somewhere secure they will hunt it down. You will be warned when your Phylactery is in peril (if those who are hunting it are near-ish) and if you don't act quickly they are able to destroy it. Leaving it a dungeon is generally a good idea, though the Necromancer's Guild works too (but they would be able to kill named NPCs in there). Note that if you are fighting bandits somewhere remote this is less likely to happen than if you do this in the middle of Windhelm while you fight Guards. Also, killing all of the hunters will end the hunt on your Phylactery, but the event can happen more than once.
Your UndeathAs a Lich you are Undead. This means spells that do more against undead, hurt you like they would an undead. {Sorry didn't realize I didn't finish this section, added it last}
As a Lich your strength seems to come from an ethereal source, so while your arms may swing a sword much the same your decaying bones cannot hold the same weight as before. Alongside your reduced ability to carry items your body is less effected by feather spells and far more effected by burdening spells. Also of importance is that not all undead are on the same side. Many liches and necromancers may still attack you, as will their servants. Vampires will despise you as well and one cannot be a Vampire and a Lich at the same time.
Lichdom: SpellcastingLiches have always been very powerful spellcasters and through embracing undeath gain a number of spells even necromancers would never be able to grasp. Moreover by accepting the chill of the grave Liches gain a bonus to frost damage (spells and enchantments).
Your SpellcastingAs a Lich you gain access to a few spells that the living would never imagine. One is the ability to convert undead, both conjured and static, to your cause for a time (though other Liches will counter-cast this and there is a limit on how many converted undead you have). Not all undead will answer to this, extremely strong creatures ie Dragon Shouting Undead will always resist this. More powerful paralytic spells are Lich specific as well as a somewhat negligible, but long lasting, DoT that has the ability to spread to other nearby enemies, applying its full duration to them. The final spell is a sustained combat hover
Society and LichdomYou are, by any metric, considerably more undead and abominable than a Vampire and with considerably less six appeal. If Society notices you, they won't be prone to niceties. Technically it is not illegal to be a Lich and as such won't attack you on sight, but only the strangest, greediest and most eccentric people will so much as let you shop with them if you're a Lich. Using spells that are Lich exclusive in front of people is a good way to have people notice what you are. Also, the longer your Phylactery is being hunted the more likely they are to out you. Finally members of the Mages' Guild (both or whichever one isn't the necromancer's guild) will know what you are immediately and will out you. As such you will be able to detect members of that Mages' Guild from a distance.
Reversing your LichdomYou do not wander into Lichdom and you do not leave it easily. You inducted undeath into yourself, the means for reversing it is hardly more honourable. Your Phylactery has to be destroyed and it can't be by you. The person who teaches you how to reverse it will be able to tell you who destroyed your phylactery or you can convince a priest to do it for you (if you haven't been outed). At that point you use a scroll to imprison them and warp them back to the person who can reverse it. You then have to absorb that person into you, using their living soul to expunge the undeath. As it takes a Phylactery to turn someone into a Lich, and you could only make one using Mannimarco's Relics you can't re-enter Lichdom.