I wanted to play an awakened bear, and decided to create my own race. I used JamesMusicus' Basic Race Feature Analysis sheet to create it, but I'd like to know how it would stack up in game. Is the unarmed strike a bit much?

Awakened Bear

Ability Score Increase. Your Strength score increases by 2 and your Constitution score increases by 1.

Age. Awakened bears, while intelligent, only live about 30 years. They tend to mature at about a year old.

Alignment. Awakened bears tend towards neutrality, though they can be fiercely loyal if they do choose a side.

Size. Awakened bears are 5-7 feet tall, and weigh 600-1000 lbs. Your size is Medium.

Speed. Your base walking speed is 30 feet. You have a swimming speed of 30 feet.

Bear Paws. You have proficiency with your claws and teeth. Your claws deal 2d6+Str slashing damage and your teeth deal 1d8+Str piercing damage. Due to your clumsy paws, you can’t use any artisan's tools, musical instruments, or weapons with the light or finesse properties.

Darkvision. You can see in dim light within 60 feet of you as if it were bright light, and in darkness as if it were dim light. You can’t discern color in darkness, only shades of gray.

Thick Fur. You have resistance to cold damage.

Brave. You have advantage on saving throws against being frightened.

Languages. You can speak, read, and write Common.

Overall, Looks Good!

Almost all of these abilities have a precedent. Damage resistance is shared by dragonborn. Proficiency in what is essentially a two handed weapon does not currently exist RAW -Dwarves and Elves are only proficient in 1 handed weapons- but I don't think this is a big deal. Any class that isn't proficient in a two handed weapon probably doesn't want to use one anyway.

Rebalancing of Bear Paws

Bear Paws. You have proficiency with your claws and teeth. Your claws
deal 2d6+Str slashing damage and your teeth deal 1d8+Str piercing
damage. Due to your clumsy paws, you can’t use any artisan's tools,
musical instruments, light weapons or finnese weapons.

I agree with Dan B that it would be helpful to parse these out for the sake of readability. I also agree that it's important to prevent the claws from being used as monk weapons. That said, you don't want to remove all flavor: if you bump the bite damage down to 1d6+Str, it will be the same as the Lizardfolk's Bite ability, which is a natural weapon that can be used for monk abilities.

It might also be good to note that these are both "natural weapons" in order to better ground them in the system.

I don't think there's any need to state how many attacks you can make with these weapons per round. None of the natural weapon abilities of RAW classes do this. No weapon description does this. Your claws are weapons, natural weapons - the rules already provide ample guidance on how weapons can be used.

Finally, it is vitally important that you specify that a claw attack requires use of both paws - otherwise, your class could use a shield in one hand and a 2d6 weapon in the other.

Smaller Concerns

The swim speed is perhaps too high, not just from a narrative perspective (a bear that can swim as fast as it can run?) but also because it is the same as the Triton swim speed, and it's probably preferable to not steal the aquatic stage from the fish people race.

From a narrative perspective, I'm curious how the bear can use any weapons at all, but that might be too strong a debuff.

In line with other spellcaster-created creatures from the MM, you might change language from "Common" to "One language known by its creator". Also, maybe "You can speak, read, and with great difficulty write in one language of your creator." would be more appropriate given the trouble using tools?

This ability:

Bear Paws. You have proficiency with your claws and teeth. Your claws deal 2d6+Str slashing damage and your teeth deal 1d8+Str piercing damage. Due to your clumsy paws, you can’t use any artisan's tools, musical instruments, light weapons or finnese weapons.

is unclear. Can you attack with both claws and teeth in the same action? If you're a fighter and you have Extra Attack, does that give both an extra claws attack and a teeth attack, or what?

If the ability lets you make two high-damage attacks for one action, then it is much too good.

Also, if the ability counts as an "unarmed strike" for purposes of monk attacks, then it is too good.

I recommend clarifying it by splitting it into two abilities:

Bear Paws. You have proficiency with your claws, and if both your hands are free then can attack with them as a weapon. This is not an unarmed strike for purposes of monk attacks. Your claws deal 2d6+Str slashing damage. Due to your clumsy paws, you can’t use any artisan's tools, musical instruments, light weapons or finesse weapons.

Bear Teeth. You have proficiency with your bite and you can attack with it as a weapon. This is not an unarmed strike for purposes of monk attacks. Your bite deals 1d8+Str piercing damage.

The "bear paws" ability is now perfectly reasonable since it's the same as holding a greatsword. The "bear teeth" thing is a little bit exploitable since it lets you make melee attacks while both your hands are full. (eg, you could have a longbow in your hands and still threaten attacks in melee) But racial features are supposed to be a little bit good, and it's unlikely to matter for a class that's focused in strength and probably can make melee attacks anyway.

Aside from that, I think what you have is reasonable.

Awakened Bears are balanced, but there are a few things to consider.

Using the Detect Balance: a 5e Homebrew Race Guide spreadsheet and my experience in the judgement of homebrew races, I'm evaluating as follows:

Ability Score Increase: ASI+3, +12 points.

Movement: 30 ft Swim Speed, +2 points.

Bear Paws: Natrual weapons 2d6, +5 points.

Bear Teeth: Natrual weapon 1d8, +3 points.

Darkvision: 60 ft Darkvision, +3 points.

Thick Fur: Cold resistance, +3 points.

Brave: Advantage on a rare roll, +1 point.

Things to consider:

What type of creature is the Awakened Bear?

If a type other than humanoid, then + 2 points. If humanoid, then -1 point.

Clumsy Bear Paws.

You can’t use any artisan's tools, musical instruments, light weapons
or finnese weapons.

This is most likely an inconsequential disadvantage at its worst it is - 2 points for the weapon use inability and -2 points for the sometimes useful Artisan tools; and -1 point for musical instrument ribbon; up to - 4 points (a frequently useful feature) for a Bard whose musical instruments can be used as a spellcasting focus (PHB 53, PHB 203).

Most likely those disadvantages will never come to play when you play an Awakened Bear consequentially I would rate them as a -1 point ribbon in most cases.

Age: Thirty years is a rather short lifespan, but mechanically inconsequential.

Bear Paws: Should be reworded to: using both of your claws

Conclusion:

The Awakened Bear amounts to 21-28 points, (or at its absolute worst 18 points, when played as a Bard), while the low side is very conditional and the high end represents a robust score.

The Detect Balance: a 5e Homebrew Race Guide recommends 24-27 points which are easily achieved if it has the humanoid sub-type. Otherwise, it is at the strong side of races, about the power-level of a Wood Elf which is still in the reasonable frame of 20-30 points.

As currently phrased, I think the Bear Paws trait is slightly underpowered, except in one very important circumstance.

The claws should obviously be compared to wielding a greatsword. Since you don't say otherwise, you would only be able to make one attack with this feature per attack action, unless you have Extra Attack, just like a greatsword. Natural weapons don't count as unarmed strikes unless a feature says they do, so they wouldn't work for a monk, just like a greatsword. They're not weapons held in hands, so they don't interact with two weapon fighting, with or without the related feat, just like a greatsword.

It's underpowered because it doesn't interact with the dueling or great weapon fighting styles of the fighter or paladin classes, and comes with all those restrictions.

However, it does work with a shield. So, for a barbarian that doesn't have to worry about the fighting styles, you're essentially adding 2 to AC. I think this is a problem. My solution would be to change it so that you can attack with both claws as a single attack that deals 2d6 damage. This means that if you're wearing a shield, you can't use that attack. You may even add wording that makes it work with the great weapon fighting style.