I'm working on a new character, not for any particular campaign, just for fun (sure we've all been there). I wanted to theme her around the concept of racing and speed. She's going to be a broom racer. Think Vanellope from Wreck it Ralph, Courier from Akudama Drive, Speed Racer, quidditch players, and so forth. 

Problem is I cant think of a good class/subclass to go with this idea, and I was curious as to what ideas you guys had about it. 

Air genasi would be a good race for you as it gives you levitation once per long rest which you could say is flying on a broom.

For class I'd say wizard or sorcerer. I'd gotta think on subclasses this is my quick answer. Spells like expeditious retreat, and longstrider could be levitation on a broom next to the ground. I'd say as you gain more spells, fly, blur, and levitate are good spells.

For wizard there isn't really any good subclasses, but I'd argue a nontraditionally played bladesinger as it gives a nice speed boost and some extra initiative. It also opens up the idea of a flying broom person who does melee, or you could just keep to ranged combat with you spells and blade song won't punish you for it.

For sorcerer I would say storm sorcerer as the 1st ability you get allows you to fly 10 feet after you cast a spell.

What are the mechanics involved in broom racing? Straight ability checks, skill checks, maybe even saves, something else? Is it a contact sport or not, what is allowed in general (can you use special abilities, maybe magic, can you tackle or otherwise attack others)? Are these brooms entirely magical or is there a mechanical component (Artificer might be useful either way)? What's the race like (just speeding around a track, is it part of a quidditch-like sport, does it involve navigating special terrain, are there obstacles or active hindrances)?  

My usual non-helpful advice for these kinds of questions is to play what seems fun to you and flavor it from there. If you actually have a broom of flying, then you can be literally anything you want.

Look for a way to get a proficiency in vehicles (air), maybe a custom background to get it as a tool proficiency. 

I’d second getting a broom of flying. It has a set speed of 50, which I read as replacing your walking speed when you use it. So you could go rogue for the bonus action dash, letting you take three moves per round. Go arcane trickster and you can cast haste on yourself for another dash.

Warlock.  Patron: The Celestial.  Protector Aasimar.  The word "Warlock" is the masculine form of the term "Witch" and if there is anything more stereotypical than a Witch flying around on a broom, I don't know what it would be.  Grab a non-magical broom.  You'll be able to get two luminous, incorporeal wings to sprout from your back, and fly around for a minute at a speed of 30 once per day.  That should be plenty for a sporting event.  You can even knock other people out of the air with a blast of eldritch energy, but that wouldn't be very sporting.

A Broom of Flying would of course, be Perfect.  That, some black tattered Robes and a Wicked Witch Style Pointy Hat

This idea sounds like a lot of fun. I would recommend designing some race rules, that might flesh out the effectiveness of character builds. For example, direct damage abilities might convert into collision values and something like Hypnotic Pattern might have a delay value. Along the lines of maintaining Concentration, you instead determine if you shake off the effect, crash, come to a dead stop, or have an ongoing penalty that affects control as you encounter other scenarios while racing. Attributes might have special functions that take an action to use, your attribute modifier reduces losing control against certain types of detrimental effects or downgrades result penalties to a lower grade on a 20 item result table. Strength gives you Bulldoze which reduces the effect of physical(non-magical) damage and barriers in regards to your racing control or table reduction. Dexterity allows Zig Zag giving a +1 avoidance or table reduction modifier to all attacks made at range that aren't Dex based, as well a providing the standard defense against Dex based saves. Constitution gives Perseverance,  your modifier determines how many times per race you can use the ability. You can use Perseverance to instantly recover from a crash or delay or, to perform a Dash as a bonus action on your turn. I would think the Chase rules might be of some interest too.

These are just some quick thoughts. Maybe a little more complicated than what you are thinking. 

Either I'm missing something or nobody yet has pointed out what I would consider the most basic and arguably important ability for a broom racer too have: feather fall. Seriously, it's your parachute in case you get knocked off your broom or get hit by a dispel magic or similar effect. Assuming your character does have a broom of flying or similarly enchanted broom, a rogue's cunning action would be great for getting to dash (double move) as a bonus action for zipping around both racetracks and battlefields. So along those lines I would suggest a wizard or sorcerer multiclassed with rogue. How much you focus on either class (aside from at least two levels in rogue to get the cunning action) and subclass options would really depend on what else you want the character to do.