Absolutist rulers claimed a monopoly of power and authority within their realms. What did they do to achieve this goal? Was any European monarch's power ever really "absolute"? If yes, which one? If no, who came closest? What happened to absolutist rulers?

The absolutists monarchs of the seventeenth and early eighteenth-centuries attempted to establish a monopoly on power and authority in their states through a number of different ways. The methods they used differed according to the preexisting structures of government and social realities in their states. In France, for example, the Bourbon monarchy had no real legislative body to contend with. He did have, however, a system of courts, controlled by nobles, that often worked at cross-purposes with the monarch. Of these so-called parlements, the one in Paris was the strongest. Additionally, nobles within cities still enjoyed old feudal privileges that they claimed made them immune to the decrees of Louis. The so-called "Sun King" attempted to rectify this by creating a bureaucratic state staffed with "nobles of the robe," lesser nobility that owed their positions to Louis himself. This was expensive and unwieldy, and truthfully Louis never really exercised anywhere near as much power as he claimed. That said, he was extraordinarily effective in promoting himself as an absolute monarch, holding court at Versailles, where he cultivated the image of a "sun king" who was the living embodiment of France itself. Louis also tried to cultivate French national unity by repealing the Edict of Nantes, which had extended toleration to French Protestants, on the grounds that it created a state within a state. So while Louis unquestionably expanded the powers of the French monarchy and the state, and was perhaps the most "absolute" of the European absolutists, his powers were always limited. In England, the Stuarts monarchy attempted to create an absolutist state modeled along French lines. But James I and especially Charles I ran up against a Parliament that was jealous of its authority, and especially unwilling to levy taxes that had not originated with its members. While Louis also had to deal with such issues, the Parliament in England was more established, and enjoyed more support from a growing middle class that resented Stuart pretensions. This issue culminated with the English Civil War, which ended with the establishment of a Puritan Commonwealth in which Oliver Cromwell ironically ruled with more power than the Stuarts had ever wielded. In eastern Europe, Prussian and especially Russian monarchs had to deal with entrenched nobles that generally opposed moves toward consolidation of power. Monarchs attempted to coopt these nobles into their power-building projects, but were only so successful. Prussian "Junkers," for example, would continue to maintain considerable influence over the monarchy because they represented the officer class in the military. Russian nobles maintained a great deal of autonomy (though they had to adopt some Western cultural trends under Peter the Great) through their control over serf and peasant populations. So overall, despite the pageantry and the rhetoric, and the fact that they did wield much more power than their predecessors, most absolutist monarchs faced real limitations.
https://www2.stetson.edu/~psteeves/classes/louisxiv.html

Absolutist rulers use ideology to help achieve their goals. For example, the ideology of the "divine rights of kings" asserted that kings ruled solely because God's will had ordained they be the monarch: this was the sole source of their legitimacy. Other bodies, such as a parliament or the judiciary, served solely at the will of the king. The king did not need their approval to be considered legitimate nor did he have to meet some standard of "good" ruler: an incompetent or despotic ruler was considered God's judgment on the people. The monarch could ignore or dissolve the parliament (or other legislature) at will. Judges also served at the king's pleasure and could be removed for countering his commands. "Divine right of kings" was also used to assert that a monarch actually owned all the land in the kingdom as well as all the goods coming into the country and could thus tax these or give land or monopolies to people at will.
Kings who ruled as absolutist took practical measures to consolidate their power, as ideology alone would not work. While these measures varied in detail, a few common characteristics reoccur. Absolutist monarchs exercised censorship over the press and artistic expression in order to quash dissent. They wooed the church or dominant religious institutions to their side. As James I of England, a king who tried to reign as absolutist, said, "no bishops, no kings." Absolutists knew they needed religious authority to buttress their own claims of divine authority. Absolutists also had a tendency to dissolve (or ignore) legislative bodies. James I's son Charles I, for example, dissolved parliament when it tried to check his power. That did not work out very successfully, as it led to civil war and Charles's beheading. Absolutist monarchs do their best to control the military and to expand the reach of a central state bureaucracy loyal only to them while surrounding themselves with a small group of powerful and loyal cronies who control the flow of money and patronage. 
While no monarch has achieved absolute power, Louis XIV of France, credited with saying "I am the state," is often singled out as one who came closest. He neutered the power of the aristocracy, classically the group in tension with the king, by building Versailles and setting up a system where, in order to gain patronage and power, nobles had to spend most of their time at that court. This way, he kept them under his control and unable to build a separate power base.