Abathur is a unique hero who does not fight his battles himself. The crafty Zerg instead places Toxic Nest on the battlefield, and empowers allies through his Symbiote.

Abathur is a really weak hero in the early stages of the game, and his lack of presence can really make his team suffer if he is not played in the correct situations. However, Abathur is also a fantastic enabler for some heroes, while also providing a lot of macro power on the map.

Toxic Nest can be used to deal damage to minions, in order to clear minion waves. Then Symbiote should be used to finish the job while gathering the XP from said minions. As every hero, Abathur can step up near XP globes to gather them, we refer to this as Body Soaking, but it is not recommended in most cases as the character can be killed in a matter of seconds due to his very low HP.
The other very important use of Toxic Nest is to provide vision for Abathur's team. Place them in paths where the enemy is prone make their rotations in order to reveal them and dismount them. Place mines near mercenaries or bosses so your team knows the enemy is or isn't taking those, etc. etc.

Symbiote can be used either to gather experience while applying map pressure, or on allied heroes to boost their performance. When Symbiote is active, Abathur puts his "Hat" on his target, and has access to new abilities: Spike Burst, Stab and Carapace.

Carapace gives a shield to the target, while also providing a small heal over time.
Spike Burst deals damage in an area around the hat, which helps Abathur clearing minion waves.
Stab is a projectile that will deal damage to the first enemy hit, which should be used on priority targets.

Abathur is a very easy hero when it comes to mechanical skill, the only difficult ability to land being his Stab; the difficulty of Abathur lies in decision making, the main decision being whether to use Symbiote to soak lanes or boost an ally hero. You will get better at decision making from a global understanding of the game, but generally know that in the early stages of the game, you should prioritize soaking XP from lanes, and only hat an ally if it would save them with the Carapace, or if you could get a kill with it.

First things first, there are other builds people like playing, but I do not recommend them if you wish to take Abathur to the highest level of play.

The Locust Strain build heavily relies on the enemy team not knowing what's going on, it also struggles getting any value against heroes that push lanes harder than Abathur in the first place, and Abathur will also lack teamfight impact with this talent path.

The Toxic Nest build is a very annoying build to play against, and it has some potential use at higher ranks, but it is so rare that you should not consider it if you aren't already extremely knowledgeable about the Hero at high rank!

We will thus focus on Symbiote builds that bring teamfight impact by boosting allies; note that Abathur does rely a lot on his allies to work, and thus his ability to carry games in Storm League is fairly limited. I personally do not recommend playing Abathur if you aren't in a party with your hat targets, but you can still find great success with it.

The build I overall recommend playing in Storm League is the General Boost Build.

At Level 1, Pressurized Glands is the talent you will pick in roughly 80% of games. Spike Burst hits a larger area and has reduced cooldown, empowering it both for waveclear and teamfight damage.
The other option that can be picked is Reinforced Carapace, which is very useful if your main Symbiote target has a low HP pool, like a Valla or Tracer for instance. In this case, it enables them to be much safer against burst damage.

At Level 4, Needlespine is the talent I recommend in Storm League as you will get value out of it in all cases. It adds significant damage to your Stab and can be used both in lane and in teamfights. It also means it adds burst damage if your Symbiote target is something like Genji, making the Ninja a much bigger threat for ganking and diving.
The other option at this tier is Adrenal Overload, which grants a huge 25% Attack Speed increase to the hero you are boosting! This talent does not help you in lane, but if you have a strong Auto-Attack hero, it is definitely the pick!

At Level 7, you should pick Networked Carapace, always! The AoE shield provides very good value in lane, empowering your minion waves to push better or denying more damage from the enemy on your minions to buy more time.
Calldown: MULE is not a bad talent, but it requires too many conditions to get proper value out of, especially in solo queue, while Networked Carapace gives more consistent value while also being useful in teamfights!
If MULE looks good to you, remember that the AoE shield would generally have applied more lane pressure which would result in not being in a bad situation where you need MULE; basically, Networked Carapace simply does the job better and more consistently.

At Level 10, you will want to pick the Ultimate Evolution which allows you to copy a hero from your team.
Knowing which character to clone can be difficult sometimes, as you might want to clone a burst mage like Jaina to try and kill the enemy team fast, or on the other hand you want to copy Johanna so the enemy simply cannot handle the amount of CC you have, but you may also want to clone a healer like Rehgar for added sustain, it's difficult to know the optimal decision sometimes.
Generally, you will want to be proactive and clone a strong Assassin, or a high damage Bruiser like Sonya. The clone lasts for a maximum of 20s, and you want to win the fight in this time span, which makes damage heroes a natural choice to copy.
You need to know that your team is going to engage, or you can ping the engage with the "on my way" ping and initiate yourself with the clone.

At Level 13, Spatial Efficiency is good if you picked Needlespine at Level 4.
If your team needs more survivability, or if you did not pick Needlespine, you should pick Soma Transference which has great synergy with Pressurized Glands at Level 1.
Soma Transference brings another way for you to heal your Symbiote target, which can really make a difference, but note that it will only work if your target is in close enough range of enemy heroes, which means they either dive or get dived on.

At Level 16, Adrenaline Boost is the default choice, as 40% movement speed is faster than Mounted speed!
This enables any hero to engage, to chase, or to get to safety (unless they are unable to move from CC).

At Level 20, Hivemind is the default pick. Having an extra Symbiote target when your entire build is based on boosting allies is indeed a huge upgrade.
Alternatively, Evolutionary Link can be picked if your clone target is a diver that needs the shield, it's great on Sonya for instance!

This build is a slight variation of the previous one.
The most notable change is that you will not use Ultimate Evolution in fights, because you want to keep your Symbiote active on a Hypercarry hero instead, a very good example being Illidan.

For this reason, and only for this reason, Evolve Monstrosity is picked.
The rest of the build keeps the same spirit where you boost an ally and enable them during fights, increasing their damage with Adrenal Overload, and their survivability with Carapace and Soma Transference.

This Hypercarry tactic can also be used on a ranged hero like Valla for instance, in which case you would still prefer the extra shield from Reinforced Carapace at Level 1, and Adrenaline Boost can also be picked for extra safety if your target is not a very mobile dive character.

With this build, you want to start using Symbiote on your target as soon as your get the Attack Speed boost at Level 4, but remember that when they are not fighting, you should still do what Abathur does, place Toxic Nest for vision and catch the Experience you can from lanes! On top of that, you should move your Monstrosity in safe spots so it takes less damage, and you can make it burrow to a different lane if you think it would get more value that way; but this remains very secondary.