After fleeing the country, Grindelwald didn't go into hiding. Instead he put focus on his quest - searching for the infamous Deathly Hallows. After much research he came across rumours which gave him a direction and put him one step closer to said task. He had heard that the renowned wandmaker, Mykew Gregorovitch, was in possession of the Elder Wand and doing experiments on it with the hope to replicate its magic properties.
One evening he broke into the wandmaker's shop, and waited for his return. When Gregorovitch entered the shop, Grindelwald stunned him with a spell, and stole the Elder Wand from him. It's unknown whether his research provided this thought of stunning the wandmaker, or if it was simply an accidental discovery, as the common assumption was that the wand's ownership can only be passed on by murdering the previous owner.
Possessing the Elder Want, the next step in Grindelwald's plan was to amass a group of loyal followers who will accompany him on his journey. He would have them launch several devastating attacks across the whole of Europe. It would catch the attention of various wizarding authorities as soon as those attacks included mass-slaughte, caused chaos and destruction. The attacks were so heavy and widespread that quite a few of them drew attention to the Muggle World, risking potential exposure of the Wizarding World and subsequently starting a war between the two worlds.
The actions were simply too disastrous for anyone to ignore them, and eventually an international wizardhunt was started and was reported in various wizarding periodicals - the Daily Prophet and the New York Ghost being some examples.
At one point the authorities had gotten a bit of luck behind them and managed to pinpoint and locate Grindelwald. They got so close, they nearly captured him - but unfortunately he was able to evade them and disappeared without leaving a trace for them to follow. The President of the Magical Congress of the United States of America (MACUSA) at that time, Seraphina Picquery, put the blame on one Heinrich Eberstadt for letting Grindelwald slip their fingers.
Unknown to Seraphina, Grindelwald was much closer than she could imagine. Using human transfiguration he had taken the identity of Percival Graves, her right-hand man who was the head of MACUSA's Department of Magical Law Enforcement. This allowed him to keep himself in the loop of MACUSA's steps, as well as use their resources and findings. He learned of mysterious attacks taking place in New York City, and had believed these to be the work of a powerful Obscurial, a wizard or witch with repressed magical energy.
An Obscurus, the manifestation of that repressed energy, is usually created when a child consciously tries repressing their talent or are forced to do so through physical or psychological abuse. It's seen as a dark and parasitic force, that manifests itself into a completely new entity with a highly violent and destructive fury.
Grindelwald had hopes that he would locate the Obscurial, and use them to his advantage to expose the Wizarding World to the Muggle World, starting a war and creating a weapon against Albus Dumbledore. He kept in continuous contact with a teenager Credence Barebone, offering him protection, believing him to be connected to the Obscurial after a vision he had received. Of course Obscurial generally cannot manage their Obscurus well and tend not to live beyond their child years, so Grindelwald had not considered that Credence himself might in fact be the Obscurial.
During an investigation of Newt Scamander, who had beasts escaped from his magical suitcase which the MACUSA believed to be the cause of the attacks, Grindelwald found the Obscurus in New Scamander's suitcase. At that time he was still in the form of Percival Graves, and had sentenced New Scamander and his friend Tina Goldstein sentenced to death in an attempt to cover his tracks and not let any further word of the Obscurus spread around.
Shortly after, the adoptive mother of Credence, Mary Lou Barebone, passed away. Her death was reportedly rather violent and her face was covered with distinct marks. Taking these news as a sign that the Obscurial is nearby, Grindelwald pushed Credence to lead him to his younger adoptive sister, Modesty. At this time he believed her to be the Obscurial, and once he was in her presence he cruelly rejected Credence as he no longer saw any need or use in having him around. This of course angered Credence, which in turn made him reveal that he was the Obscurial.
While Grindelwald was shocked by the discovery, his plans had not changed. He attempted to recruit Credence, to join his ranks, by telling him that he had nothing to fear anymore. Credence did not take this very well, and instead ended up rampaging through the city and wreaking havoc.
"We've lived in the shadows for far too long." ~ Gelert Grindelwald