RAW
No damage dice => No dice to double
Sucks, but them's the rules. So a critical hit with an unarmed strike does nothing more than 1 damage, excluding modifiers. (I'll be excluding those for the rest of the answer) The tables in the player's handbook doesn't list a "1d1", just 1 damage. "1" is not a dice roll, it's a number.
Of course, the designers do seem intent on making unarmed strikes do 1 damage. I doubt it's a typo, especially considering the basic rules have been updated (it's on v0.2). They even supplied a feat (tavern brawler in PHB) and a class (monk) which increases the unarmed strike damage from 1+mod to some actual dice (like a 1d4). If they never intended the damage of unarmed strikes to be so lame, why would they provide that feat and that class?
As a side note: a critical hit from any weapon causes two failed death saving throws... including unarmed strikes. This is possibly the only time they are on par with more traditional weapons.
Possible RAI and RAF; The Spirit of Critical Hits
Critical hits seem to be meant to do more damage than a normal hit. This is reflected in the doubling of an attacks damage dice. It makes you generally do more damage than normal, although there is a slim chance that you don't do as much on that strike. You can see the odds for some weapon dice values here.
If you're looking to extend this logic to critical hits, I suggest instantiating a house rule for upping it to a 1d4 or supply 2 damage on a critical hit. This could be a fun rule, and it may alleviate the disappointment your players feel while getting that critical hit while hitting something with their... fists? ("Unarmed Strike" is really vague!)
However, it does seem that the designers really did want unarmed strikes to not hit hard unless you've had some experience. Once again, this is because of the Tavern Brawler feat in PHB and the monk class. Is this because they want people to use weapons unless they're a monk or a tavern brawler? Were they afraid of things like the grapplemancer builds coming into 5e? Hard to say. You could ask wizards of the coast by emailing sageadvice@wizards.com.
A Note
5e greatly acknowledges the importance of house rules in D&D culture, so I would not stick with the RAW reading of this. In that article linked above, written by Jeremy Crawford, one of the two lead designers for 5e, says:
We expect DMs to depart from the rules when running a particular campaign or when seeking the greatest happiness for a certain group of players.
The RAW just is not fun and takes the joy out of a critical hit. Come up with a rule that seems right (doubling to 2 damage, or going to 1d4 damage, or something else) for your group. In the words of Jeremy Crawford:
Fifth edition now belongs to the thousands of groups playing it.
Modify 5e to have fun in your group!
Unarmed attacks can crit, but do not do double damage
Can an unarmed strike crit? - Yes
If you roll a 20 on an unarmed strike, it is considered a critical hit.
This means that it will automatically hit regardless of the opponent's AC and has the potential to cause 2 failed death saving throws from a creature at 0 HP.
However, a critical hit does not get to increase its damage from a critical hit.
Do unarmed strikes get double damage on a critical hit? - No
Critical hits only double damage that you roll. Unarmed strikes do not have any damage rolled. So, RAW, critical hits do not increase the damage of unarmed strikes. See this Q&A for further discussion of this.
Jeremy Crawford agrees and has (unofficially) clarified:
An unarmed strike deals 1 + Str. modifier damage, even on a critical hit.
This ruling, besides being in line with the literal reading of the RAW, is likely made to keep the rules of what damage to multiply from a critical (only rolled damage) consistent. If you have to start carving out exceptions for every static damage effect that doesn't roll damage but you think might kind of be like a roll, applying the critical hit rule could get complicated and cumbersome. Thus the simple, easy-to-apply ruling we get here.1
Does the described attack cause failed Death Saving Throws? - No
It is possible for an attack to do 0 damage
Jeremy Crawford has (unofficially) clarified:
There is not a damage minimum of 1, so it is possible to deal 0 damage with an attack.
So you are indeed calculated the damage for that attack correctly.
0 damage does not count as damage
Jeremy Crawford has (unofficially) clarified:
Taking 0 damage is the same as taking no damage.
No damage means no failed death saving throws
If an Unconscious character takes damage while at 0 HP, they automatically fail one death saving throw, or 2 death saves if the damage is from a critical hit.
If a creature takes damage while at 0 hp, then they will fail one saving throw (or two for a crit). If no damage is taken then no death saving throws are failed. The critical does not change this because damage is a prerequisite in order for any death saving throws to be failed in this way.
So the character in this case does not fail any death saving throws as a result of the attack
1 - Note that I've played in and run games where I've allowed unarmed strikes to double damage as a house rule, and it seems to have had only positive effects (if any, since it came up extremely rarely). But it was always made very clear that this exception was being made for only this one thing. If you did so at your table, I'd guess you wouldn't have any issues either. The only thing to watch out for is confusion I'd say, because the rule that only rolled damage is rerolled is a handy one, and breaking that for unarmed strikes might undermine that. I've only done this house rule with experienced players, so keep that in mind.
Best Answer
You add your Strength bonus to any melee weapon damage you do unless a rule specifically states otherwise.
As per the rules on what Strength does for you:
In the original printings of the rules, unarmed strikes were a weapon with an entry on the PHB's table of weapons. However, they were removed from this table by errata, and instead the rules for melee attacks state that:
This means that unarmed strikes are not, technically, an "attack with a melee weapon", but they are a "melee weapon attack". The following rules about damage rolls unfortunately also use the "attack with a weapon" wording:
Which strictly rules out the unarmed strike. However, Sage Advice rules that when you make a melee weapon attack, by default you add your strength to the attack roll and to damage. This general rule is always in effect unless it is explicitly overruled by something else:
The Aarakocra's unarmed strike doesn't explicitly include the Strength modifier, but it also doesn't explicitly exclude it. Additionally, other races with modified unarmed strikes published in later material, such as the Tabaxi, do specify that the unarmed strike damage still includes the character's strength modifier, and the implementation of Aarakocra on the official companion site dndbeyond.com includes the strength modifier to damage.
Though the actual letter of the published rules is not as clear as it could be, the most consistent interpretation of the material is that Aarakocra should still include their strength modifier on unarmed strikes, and it is an oversight that such was not included in their description. The Elemental Evil Player's Companion has never had errata published, but one imagines if it did this would have been corrected so as to be unambiguous.