Reports are very conflicted. The police staked him out on at least one occasion, but the Empress Alexandra didn't want to hear of anything bad said about him. Nicholas seemed to go along with her wishes. After some accounts of ribaldry reached the palace, Rasputin begged the empress for forgiveness, saying that he was a sinful, imperfect man. The empress seemed to accept his apology.

Rasputin was originally sent for when the Empress, desperate to have her son Alexei healed from his illness, allowed the starets (mystical man of God) to treat her son.

Overjoyed by Rasputin's apparent healing of the tsarverich, the weak little boy on whom the dynasty relied upon, Alexandra showered him with gifts. He began wearing a silk blouse, had a personal secretary, and was given a car. He also began making policy suggestions to Alexandra, who was filling in for Nicholas while he was away at the front, like some kind of ancient tsar-warrior leading his troops. In real life he could do little - he had only ever obtained the rank of colonel in the army, IIRC.

So was Rasputin a true, yet flawed holy man or a mere charlatan? This would depend on who you ask. Most historians do not, of course, view him positively. Few would doubt that the tsarervich did seem to get better after Rasputin saw him. (Rasputin had said that the boy should not be disturbed by doctors. Coincidentally or not, we now know that sufferers of hemophilia, young Alexei's condition, are sometimes best left alone).

According to Prince Felix Yusopov, a close relation to the tsar and possible the richest man in tsar Russia, Rasputin had once told him, boasting of his influence on the empress: "I'll have you made a minister if you like" or words to that effect. However, Yusupov, who together with other nobles went on to murder Rasputin, was known for embellishing the truth, to put it mildly.

Perhaps we'll never truly know.

(Source: Nicholas and Alexandra, by Robert K. Massie)

I'm going to argue slightly against the other answer(s) that has been given and argue that Rasputin's reported influence over the Romanov Dynasty has been largely overstated.

Ultimately the question of Rasputin's motives are hard to pull out of history. He clearly was a charismatic man who held beliefs of some kind. The man was Nicholas II's personal confessor and his supposed healing of Alexei's hemophilia definitely allowed him access to larger amounts of influence within the royal family. I don't think he ever really wanted political power, or to significantly mislead the royal family. It seems to me that he was able to work his way into their good graces by being the right person in the right place and at the right time.

Now as to his influence on politics in late Tsarist Russia, as I stated before I think his own personal influence has been overstated immensely. Perhaps this is best evidenced by how little changed in Russian politics between his rise to prominence and his assassination. The image I think that is most often presented in regards to Rasputin is that through his healing of Alexei he was able to grow close to the Tsarina. Close enough that rumors of adultery between the two spread. This combined with the perception (one that both contemporaries and historians of today hold) that Nicholas II was heavily influenced by his wife Alexandra (who herself was viewed with distrust by the Russian public due to her German heritage) have lead to people arguing that in turn Rasputin was able to influence Nicholas heavily during WWI.

Nicholas II was a ruler who utilized what Sarah Badock has termed "highly personalized power". This meant that it was incredibly hard for the Russian government to properly prepare for war in 1914 as Nicholas retained and spent vast quantities of time on more ceremonial aspects of Tsarist rule that had been tradition for centuries and control was largely rooted in his person as opposed to ministers and ministries. Because of this Russian began war poorly quipped in terms of artillery shells, uniforms as well as poor leadership organization. In turn the low moral of Russia's peasant soldiers all meant that Russia's war effort went poorly and this culminated in 1915 with the 'Great Retreat'. Nicholas' personal style of ruler ship meant that when Nicholas made himself supreme commander of Russia's army in 1915 he, more than ever, linked the failings of his autocratic state with Russia's military failings throughout WWI further reducing confidence in his government. His decision to appoint himself as supreme commander was both productive and unproductive. In some ways the very fact that his appointment increased the linkages between the autocratic state and the army meant that there was less rivalry or conflict between these institutions, yet this very linkage would prove to be very damaging.

Paul Miliukov, a historian and liberal politician, famously took to the floor of the Duma to deride the government's handling and planning of the war effort saying, "glupost’ ili izmena", was it stupidity or treason? This sentiment was widely shared by the Russian populace, widespread derision for the apparent incompetence that had lead to the numerous failings in Russia's preparedness for war. Miliukov's statement also shows how far he and other liberal politicians had shifted in their support for the government. Initially the war proved to be something for a divided society and polity to rally around. Yet as Russia continued to suffer setbacks Nicholas' personal reputation continued to fall, as did faith in his government and the overall idea of Tsarist rule, all of which were key to his ability to effectively wield this "personalized power". Perhaps Rasputin's greatest influence was to further damage the reputation or public image of Nicholas and the royal family (which was so key to Nicholas) more than any actual political decision making by the man himself.

The Russian government's failing in war exacerbated political divides within the country. The shortages of food and other good for civilians gave rise to 'voluntary organizations' the experience of which taught local Russian government and civilians how to rule efficiently and effectively themselves and further exposed the failings of the state. In turn Nicholas was unable to work with the Russian Duma despite their overwhelming outpouring of support for the war effort and general patriotism which both further emphasized the problems of tsarist rule and the Tsar's increasing isolation on the front. Society's experience of mobilizing the Russian economy for war often without state support fostered the opinion that the Tsar should work with society (through the Duma) more and also exposed the failings of true autocratic rule. Rasputin's role in this was minimal if it existed at all and he was more a representative of some of these problems than the cause of them.

I'm sorry to write so much about other people besides Rasputin but this pervasive notion of Rasputin wielding significant influence and having really affected anything at all is one of my biggest pet peeves in Russian history.

Sources

Late Imperial Russia - Problems and Prospects - Edited by Ian Thatcher

Russia in the Age of Modernization and Revolution, 1881-1917 by Hans Rogger

Edit

I suppose, having re-read this, that I didn't explicitly answer OPs question. I suppose I would emphasize the struggle that reconstructing Rasputin's personal goals/drive would be. I would also argue that it is ultimately (an I mean this with no disrespect) a bad historical question. I don't think we gain any interesting insights into late Tsarist Russia by asking what Rasputin wanted. Asking historical questions is hard (see here) and I applaud anyone who takes the time to do so. I suppose I was trying to take OPs question and his desire for information and hopefully help steer him in better directions of inquiry! I feel like that’s half the idea of this sub.

What did he do to 'heal' Alexei?