Beatty became First Sea Lord because he was a skilled political operator, able to gain significant support within society. He was able to spin his war record, passing off his failings on scapegoats and emphasising his good qualities. He had the support of the press, and to a certain extent, Britain's political establishment. He was also a fairly natural choice, being one of the most senior leaders in the Royal Navy. His predecessor in command of the Grand Fleet had also become First Sea Lord, and it would have been hard for him to be overlooked for the same honour.
Beatty had spent a long time cultivating both a public persona as a hero and deep relationships with his seniors in and out of the navy. In his first command, he gained a reputation for heroism, throwing an unexploded shell overboard from his gunboat on the upper Nile. During the campaign, he also significantly impressed the army commander General Kitchener, receiving an accelerated promotion and a decoration. His reputation for heroism was enhanced further during his actions ashore during the Boxer Rebellion in 1901. Here, he played a major role leading in the British naval brigade, despite two major wounds to his wrist and arm. His role in the forefront of the fighting resulted in another promotion, this time to captain. As a captain, he impressed his superiors in the Atlantic Fleet through his skill and energy, particularly the well-connected Prince Louis of Battenberg. His wife, the American heiress Ethel Tree, gave him social connections, both through her wealth and her personal ambition. This, taken together, meant that he earned early promotion to the rank of Rear Admiral, despite not having had the required amount of sea time. As an Admiral, he had the fortune of being appointed, in 1911, as the naval secretary to the new First Sea Lord, Winston Churchill. This gave him access to senior politicians - in 1912, Churchill and Beatty made a Mediterranean cruise on the Admiralty yacht with Prime Minister Asquith, Kitchener and other officials. This experience helped him to secure future positions, most notably as commander of the Battlecruiser Squadron. Here, he, in part chose for his staff officers he felt might further his career. His flag lieutenant, Ralph Seymour, was chosen not for his signals background (which was limited), but because of his aristocratic connections.
Even his wartime experiences could be spun to further the legend. The British public was crying out for a naval victory over the Germans, which Jutland did not represent. Beatty was able to argue that his aggression at the battle was the key to victory, one frustrated by Jellicoe's timidity. This further raised his public profile. At the end of the war, with Beatty no longer needed to command the Grand Fleet, he was free to step up to higher roles. A major press campaign began to pressure for Beatty to become First Sea Lord. This was might have been orchestrated directly by Beatty, or by his friends in the press; it was particularly popular in the papers of Lord Northcliffe, which had been consistently pro-Beatty throughout the war. This public pressure made it almost untenable for any other officer to be First Sea Lord. In May 1919, Churchill wrote to Lloyd George on the topic, stating that the current First Sea Lord, Admiral Wemyss was "in a weak position in his own profession and far overshadowed by Beatty", meaning that "sooner or later Beatty will have to replace Wemyss". This was doubly true because, in 1918, the First Lord of the Admiralty, Eric Geddes, had promised that Beatty would become First Sea Lord when Wemyss stepped down. At the time, this was thought to be a relatively short time in the future, with Wemyss being promised the sinecure of the governorship of Malta - but this fell through at Churchill's insistence, with Wemyss holding on at the Admiralty instead. Instead, Beatty had to wait for public pressure on the Admiralty to win out later in 1919.
Beatty's failings during the war did not come back to haunt him, largely because many of them could be blamed on others. The miscommunications between 5th Battle Squadron and the Battlecruiser Fleet (BCF) at Jutland, which put 5th Battle Squadron (5th BS) into a very dangerous position, could easily be blamed on Seymour. Beatty was responsible for establishing a good working relationship between the BCF and the attached 5th BS before the battle. He failed to do so, meaning that 5th BS was unaware of how the BCF worked and communicated. Beatty was responsible for drafting unambiguous orders, which he again failed to do at Jutland. While Seymour certainly made a mistake when transmitting the orders to 5th BS, there were more significant failings made further up the chain by Beatty. Despite this, Beatty ostracised Seymour post-war, and his accounts of the battle heaped oprobrium on him. Similarly, the loss of the battlecruisers Queen Mary and Indefatigable under his command at the battle could be blamed on others. Their loss was not the result of his aggression; they were lost due to systematic factors around propellant manufacture and storage that I won't describe here, but that Beatty's actions before the battle did play into. Beatty (and Jellicoe) initially recognised these structural factors and their effects in internal communications. However, he soon shifted to arguing that the issues the ships faced were the result of their poor protection. In a letter to Jellicoe of 14th July 1916, he argued that "our methods of ship construction are seriously at fault"; in other words, the losses at Jutland were not his fault, but the fault of the ship designers. Beatty, along with Jellicoe, pushed for the argument that the British battlecruisers were too lightly armoured. This absolved both of fault, and cast a long shadow over the fleet's ship design in future. Other issues were covered up to allow Beatty to portray himself as the 'hero' of Jutland. Most famously, Beatty tried to cover up a 360^o turn made by the BCF later in the battle, which let the action move away from him. Covering this up in the published records let him portray himself in a more capable, aggressive light.
The scuttling of the German fleet at Scapa Flow, though, cannot be put at his feet. The Grand Fleet had split up in April 1919, and Beatty was no longer in a command role as a result. This was two months before the scuttling. The Atlantic Fleet, which replaced the Grand Fleet as the main British force at Scapa, was under the command of Admiral Fremantle, while the base itself was commanded by Rear Admiral Prendergast. Beatty could do nothing to stop the scuttling, especially since he was in London schmoozing with royalty, having just returned from a tour of the Mediterranean.