Ok so I have heard that a few people are still confused on how Eobard is still alive and his storyline so I thought I would just make this post to help clear some misconceptions. This will be from the moment he was born. Spoilers for Flash Season's 1, 2, 3 and Legends of Tomorrow Season 2.
Eobard Thawne was originally born in the year 2151 and once he became an adult he studied the Flash in order to become just like him. Once he got Super Speed he also gained access to the Speed Force and so he looked through his own timestream to see his own future and what he becomes. He see's that his destiny is to become the Flash's greatest enemy and this pretty much drove him mad.
After many battles with the Flash, Eobard Thawne discovered the Flash's real name (Barry Allen) and with this knowledge he decided to go back in time and kill Barry as a child so that he could live the rest of his days without the existence of the Flash. When he travelled back in time he was confronted by Barry from the year 2024 and the two of them fought. In the end the Flash won and saved his younger self and so Eobard killed Barry's Mom in the hopes that this trauma would stop Barry from ever having the willpower to become a Superhero.
After this fight, Eobard attempted to travel Back to the Future but he couldn't as his connection to the Speed Force had been damaged permanently after his fight with Barry. Eobard was now trapped in the past and realised the only way he could time travel back to the future was to create the Flash and use him to create a time portal in which he could travel through.
He then killed a major scientist of the decade Harrison Wells and stole his face and identity to make his particle accelerator sooner than what the current timeline foretold. Over the next 14 years, Eobard watched over Barry, kept him safe and worked on building Star Labs and the Particle Accelerator that would give Barry his Super Speed.
After the particle accelerator explosion Barry is granted Super Speed and works with Eobard, Caitlin and Cisco to keep Central City safe from evil Meta-Human's.
Eventually Team Flash discover's Eobard's real identity and so Eobard proposes a stalemate. The deal is that Eobard will give Barry the secret to time travel on the condition that Eobard can return home, in exchange Barry can go back in time and save his mother. Barry has a last minute change of heart and fights Eobard. Barry is unable to defeat the Reverse Flash so Eddie Thawne (Eobard's ancestor) commit's suicide to stop him. Eobard is then erased from existence and Eddie dies.
Past Version of Eobard Thawne:
7 months after the defeat of Eobard Thawne, the team encounters a past version of the Reverse Flash and this raised many questions as to how he still exists. This is possible because Eobard Thawne in 2016 hasn't NOT been born yet, the 2 places Eobard cannot visit in time is the moment he is erased from existence and any point after he was meant to have been concieved. This means that Eobard will truly cease to exist after the date that he hasn't been born (If you get what I mean) and he still died the day Eddie committed suicide.
In may of 2016 Barry time travelled back to the night his mother died and saved her. After he did this he took the Eobard Thawne from that time and time travelled back to the NEW 2016 that was dubbed Flashpoint. The timeline has been changed once again and this means that the Eobard that Barry kidnapped is technically a Time Remanant. After Barry worked with Eobard to restore the timeline Eobard left with his ability to time travel still intact.
Because this version of Eobard escaped he was never around at the SPECIFIC MOMENT that Eobard Thawne of 2015 was erased and he is not in the timeline where he hasn't been born, this means that Eobard has escaped his own non existence.
The Speed Force is a living entity. Think of the every moment of the timeline as a classroom and the Speed Force is the hallway with the doors. The Speed Force is a living time vortex and it becomes very unhappy when people alter the timeline and was even more unhappy when it learned that Eobard had escaped his own nonexistence. In order to resote balance to the timeline the Speed Force needed a type of Bounty Hunter that could hunt down rogue Speedsters such as Eobard Thawne, the entity they used is the Black Flash which was formely Hunter Zolomon.
After being persued by the Black Flash, Eobard came up with a plan to stop the Speed Force from killing him. He heard of an artifact called the Spear of Destiny, this is a device that can be used to alter reality and this would give him the ability to defend himself against the Speed Force. He recruited Malcolm Merlyn and Damien Darhk because they are both very skilled assassin's and he needed them to do the jobs that Eobard could not be around for. The problem he now faces is the Legend's of Tomorrow.