It's the resident joke mentioned even by Fred in his pirate radio programme:
"He can run faster than Severus Snape confronted with shampoo"
-Harry Potter and the Deathly Hallows
Since he was a student at Hogwarts, he's been mocked for his unkempt hair and for a teenager that can be quite painful. Why didn't he brew a potion to fix it? Severus Snape is supposed to be good at this stuff...
It's been well addressed that Snape was somewhat of an awkward kid with amazing potion-brewing abilities. Here's my take on why his hair was so greasy:
In Deathly Hallows, chapter 33, "The Prince's Tale," we get to see all of Snape's (important) memories, all the way back to when he was a child. The first description of him we get from these earliest memories isn't very pleasant:
"His black hair was overlong and his clothes were so mismatched that it looked deliberate: too short jeans, a shabby, overlarge coat that might have belonged to a grown man, an odd smocklike shirt."
Honestly, if I heard that without knowing it was about Snape, it would sound like the description was referring to a Weasley! Hand-me-downs from varying sources sounds like Snape's family was very poor. He had very little control over the quality of his clothing. They probably didn't have any money for anything to help his hair as a child.
Moving on to his teenage years, I doubt the family found any extra money still. Snape has been developing his potion skills, and he would definitely know how to fix his hair. My guess is he tried it once or twice, but of course it would wear off after a time. He most likely didn't have much money for so many ingredients as to keep a constant supply of hair-helping potion. He had plenty of practice with potion making though, which would mean he probably had access to the Hogwarts potion ingredients room, but I doubt he would have been able to take the ingredients for that much personal use of hair keeping. Therefore, he kept his greasy hair for his teenage years.
Fast forward to when he holds a steady job as the Hogwarts potions teacher, with enough money to do what he wants and all of the ingredients in his office. Even though he had it all accessible to him, why would he? The love of his life Lily was dead, which looking at his past was the only noble thing he cared about (the only thing else he cared about was the Dark Arts and I don't think getting rid of greasy hair is on the top of that list). He did not care for what his students thought about his hair, because he of course could intimidate them through his position as teacher; naturally he was very rude to most students and fixing his greasy hair wouldn't change how the students viewed him.
So basically, the only reason I can think he would fix his greasy hair would be for Lily, and the only time he had the resources to do so was after she was gone. He had no reason to as a teacher and no ability to as a child.
J.K. Rowling addressed this in an interview back in 2007. Her answer is short but effective...
Hannah: Why was Snape so badly groomed?
J.K. Rowling: Hmm. Good question. Poor eyesight? Did he look in the mirror and believe he was gorgeous as he was?
J.K. Rowling: I think it more likely that he valued other qualities in himself!
I like the above answer, but my personal fun theory is that Snape refused to use the most well-known hair potion (and the only example I can think of), which was created by....
Fleamont Potter, father of James and grandfather of Harry.
Henry’s son was called Fleamont Potter. Fleamont was so called because it was the dying wish of Henry’s mother that he perpetuate her maiden name, which would otherwise die out. He bore the burden remarkably well; indeed, he always attributed his dexterity at dueling to the number of times he had to fight people at Hogwarts after they had made fun of his name. It was Fleamont who took the family gold and quadrupled it, by creating magical Sleekeazy’s Hair Potion ( ‘two drops tames even the most bothersome barnet’ ). He sold the company at a vast profit when he retired, but no amount of riches could compensate him or his wife Euphemia for their childlessness. They had quite given up hope of a son or daughter when, to their shock and surprise, Euphemia found that she was pregnant and their beloved boy, James, was born.
There may be a practical reason for his unkempt hair and lack of grooming in general. Namely that he is the Potions teacher.
He teaches in an enclosed space, underground. He teaches students to brew potions. Potions emit smoke and various noxious fumes in the best of circumstances. When brewed by students, many of whom aren't particularly interested or talented in mixology, the results tend to be more vile and explosive.
Unlike the students, who have one class each day, Snape has to teach multiple classes and thus is exposed to more, and more varied, effusions.
His greasy hair could be his acknowledgement that it is going to get messed up anyway. It could be the result of the side effects of the fumes it is constantly exposed to. Or it could even be a defense to keep the worst of it from reaching his follicles or scalp.
Another thing to remember is that 99% of the book series is told from Harry Potter's point of view; his thoughts, reactions, perceptions, etc... This is not to say that Harry is an entirely unreliable narrator of course, but one could imagine that we are technically really only seeing the other characters through Harry's eyes. Yes, his observations are generally backed up by the supporting cast, (ie - nobody in Gryffindor likes Snape)
As a child I was a lot like Snape: Bright, abused at home, bullied at school, with abysmal hygiene.
At some point I simply decided that I wasn't going to wash. The other kids tried to change this with varying degrees of pestering and bullying, but because I was already alienated from them, I couldn't have cared less how they felt, and I had no interest in being more like the people who treated me like crap.
After I grew up and moved out, I started bathing regularly, so I am less like Snape there.
Speaking from personal experience: When you are a bit of a social outcast, and you don't really care what others think of you (for the most part), then you are liable to let some of your personal grooming lapse below what others consider minimal standard. (Not enough to risk people thinking you are diseased, mind you...)
One possible reason is that is the byproduct of some sort of protection. Advanced potions can be very sensitive to added ingredients. The grease might be there to hold any loose hairs until he could remove them outside of the lab.
As to why a spell could not do the same thing there could be interference issues. Therefore a non magical grease.
