Are you TIRED of PCs just being straight-shooters, clean-whistles, or even a (dare I say it) GOODY-TWO SHOES?! 
Do you want to create a PC with a flaw that actually COULD KILL YOU?! 
Perhaps you'd like your character to have some kind of story-driven bend about them that in no way, shape, or form is beneficial to you because, like me, you either enjoy a challenge or you're a masochist (I'm a bit of both).
I present to you, a simple variant rule: The Addiction Meter! 
Having an addiction is entirely optional, though you may want to let your DM know beforehand if having your PC use this rule is okay, especially if it's story-related. It is represented AS A FLAW so if you induldge into your flaw, the DM may give you inspiration if deemed appropriate to do so.
How It Works:
There are two types of addictions here: Physical and Mental. A physical addiction is usually easier to manage and sate, such as an addiction to alcohol, tobacco (pipeweed in LotR D&D players), or my character's favorite, healing potions.
Physical Addictions:
Physical addiction is based upon the PC's Constitution. Your PC's Addiction Meter (AM) is 4 + CON modifier. At the end of a long rest, the DM rolls a d4 and asks you to make a DC 15 CON saving throw to reduce the result by half (minimum 1). You tally this number into your meter, which starts at 0 and goes up, but never goes past the maximum.
Let's take Xion here with a CON of 14. His AM is 6. He meter starts at 0 and is perfectly fine for several days as it slowly works its way up to reach 6. Nothing adverse happens to him as his meter rises. Though, one morning, he wakes up with a withdrawal because his meter had reached up to 6.
At this point, the DM asks Xion to make a DC 10 CON saving throw. Should Xion succeed, nothing bad happens for the entire day. However, the next morning, the DC increases by 2. Once he fails, he suffers ONE LEVEL of EXHAUSTION (p. 291 PHB). He repeats this process every day and the success rate increases the DC by 2 (failure does not increase it) until he can sate his addiction.
Mental Addictions:
Mental addictions is based upon the PC's Wisdom. Your PC's AM is 4 + WIS modifier. You follow all the same rules as you would with a physical addiction, but the PC makes a DC 15 WIS saving throw instead of CON.
A mental addiction could represent something like being overly superstitious and paranoid, etc. You could have a PC that believes that demons are going to get him and if he doesn't pour holy water on himself once in awhile, they will invade and DRIVE HIM MAD. Though, in truth, there are no demons out to get him and it's all in his head, but he could suffer from madness effects... 
A PC with a mental addiction does not die when reaching level 6 exhaustion. Instead, they go comatose/insane. Or worse, they could die via suicide.
How To Reduce The Meter
If a PC wants to sate their addiction, they merely do so via the appropriate action, depending on what it is. Xion is addicted to healing potions, so drinking one is fine. Whenever a PC sates their addiction, the meter resets back down to 0 and all levels of exhaustion gained from this vanish. If the PC had exhaustion levels from other sources, such as being hungry, they remain.
Note to DM and Player Alike:
Work with each other and agree upon an appropriate addiction. Being addicted to food, water, and air is silly as it is a basic need that is usually easy to fulfill and as such, poses no threat to the PC (unless it's Dark Sun). Being addicted to a narcotic, or even a beneficial potion could be interesting. Something that costs a little money and is somewhat available is usually good. Just don't make it something incredibly rare or TOO dangerous (potions of longevity or drinking the poison of a wyvern). Though, trying to build up an immunity to iocaine powder is perfectly appropriate.
For those who would rather have a mental addiction, you're going to have to get creative (which is what this game is kind of all about).
Warlocks could have a much easier time roleplaying a mental addiction. Especially if it's the Great Old Ones. 
Thoughts? Comments? This is complete and utter garbage? 
Sounds nice.
But I don't thi anyone wants to complicate the game further. It's already tough enough to dungeon crawl and get through encounter after encounter, preparing for the BBEG.
I personally like your sort of "role" playing with these so-called addictions but these days everyone wants combat ("roll" playing)
I know. 
This is just an optinal rule. You can either take it or leave it. Trust me, I love combat too, but I also love a healthy mix of role-playing on the side. I don't like creating super heroes so to speak and I aim to have them have some kind of thing wrong with them that they must overcome. That's what makes my hero feel heroic to me.
This is for people who do want to complicate the game further lol. Though, it's not really that complicated. If anything, it's handled almost like a magic item that has charges on it. Almost.
I agree. It's not reallly a complicated little thing at all.
I like it in fact.
But I know my players would want something that hinders them other than monsters, traps and the like.
Maybe your table is different, and a bit more mature fir something like this.
I remember running Ravenloft in 3e and there were the "Mists" in the land which I created a 10% miss chance (if outdoors in the Mists) and that they didn't mind because it meshed with the setting. In a similar fashion I envision your little addiction mechanic to work well in a game with high involvement with drugs, rogue underworld dealing, and a "Gotham city" type setting.
Good luck!
I actually have a model for physical addiction. I'm gonna put it here not to shoot yours down, but to give it some compare and contast.
Firstly, addictions can be acquired in addition to starting with one (I have a red dragon who rolled poorly and is addicted to an opium-like substance.)
How to get addicted: Addiction Rating (AR) is given to any addictive substances. Every time a substance is used roll a CON save. If the rol is lower than the AR the player becomes addicted. If another of the same substance is used within a certain amount of time after using the first, or the Addiction Threshold (AT) the save DC is increased by the AR.
Coping with addiction: If you become addicted to a substance the effects are felt after one AT goes by without using the substance. Each item has a withdrawal effect that works opposite to its beneficial effect. Many have effects that get worse over time. Every time you reach the AT you roll a CON save (DC=10+AR). On a successfu save you suffer no ill effects and the DC increases by 1. On a failed save your withdrawl effects increase and the save DC becomes 10+AR. You lose your addiction on 3 successful saves.
As a note, my campaign has a drug-ring where addictive items can be purchased for much cheaper than potions that do the same thing. Potions are well-regulated for quality, but expensive. Three such items are listed below.
Fixer (knockoff potion of healing)
10gp
AR: 4
AT: 12 hours
Heals 2d4+2 hp.
Withdrawl Effects:
level 1: Max hp drops by 1d4 (minimum of 1).
level 2: Max hp drops by 3d4 (minimum of 1).
level 3: Max hp drops by 5d4 (minimum of 1).
Slinger
30gp
AR: 6
AT: 24 hours
Regains 3 level 1 spell slots or 1 level 3 spell slot.
Withdrawl Effects:
level 1: Uses up 1 level 1 spell slot per day.
level 2: Uses up 2 level 1 spell slots per day.
level 3: Uses up 3 level 1 spell slots and 1 level 2 spell slot per day.
level 4: Uses up 4 level 1 spell slots, 2 level 2 spell slots, and 1 level 3 spell slot per day.
level 5: Same as level 4 and all damaging spells do half damage.
Hearty
50gp
AR: 5
AT: 12 hours
You may use one hit die as a bonus action on your turn. This effect wears off after three hit dice are used or one hour passes.
Withdrawl Effects:
Level 1: Gain 1 level of exhaustion. You cannot lose levels of exhaustion on a rest.
In the example of Hearty, since the effect is worded as 'gain' that means on each failed save the level 1 effects are felt again since exhaustion stacks.
It should be noted that my campaign money is tight, so buying drugs vs. buying potions is a difficult decision.
So your system is about becoming addicted while mine is for a PC who has been addicted to whatever for a few years or so and is now going on adventures and whatnot. I get it.
I noticed something. To me, the AR may be a squeek too low. Instead of 4 and 5, I'd go straight to 6 in the event someone decides to pump CON to max (if they somehow manage to do that). They wouldn't become addicted.
Also, don't forget about classes that add a proficiency bonus to CON such as fighter and barbarian. They'd be able to laugh off any threat of addiction.
I think it's possible to have roleplay that is supported by gameplay where you gain points for it. It doesn't need to be 'hey, take a negative...and maybe you'll get a positive for it at some point. Maybe'
I don't know why these idea always seem to go counter against good stat using gameplay. Thespian ideal, I guess.