Well, let me first say that I love this question. Not only is it a very interesting one, but it just so happens that the Komnenoi are one of my specialties. I'm actually going to take a little bit of time out of work to write this because I'm very excited to answer. Note: I'll probably edit this a bit when I get home.

To start, let me first refresh your memory as to the events that led up to the rise of Alexios Komnenos:

In AD 1025, Emperor Basil II died childless, leaving the Byzantine Empire in the hands of his elderly brother Constantine VIII. Basil's reign had been a glorious one, the pinnacle of over a century of continuous and successful expansion of the Empire under the consistently effective leadership of the Macedonians, which began with Emperor Basil I's accession in AD 867. Basil II's reign had seem the expulsion and, in some cases, the essential annihilation of many of Byzantium's enemies, and better yet, it had been done while the coffers were being filled. By the end of Basil's reign, the Empire controlled most of Southern Italy, the Balkans, Anatolia, and even extended as far as Armenia and Antiochan Syria, and ruled over a population of some 20 million souls. The military had been completely restored, perhaps even to levels rivalling that of the old Empire under Justinian, and the borders were well-secured by a new and effective network of forts along the mountains of Armenia and along the Danube. The City of Constantinople, too, returned to population levels that probably rivalled that of Justinian's Empire and brought in huge sums of coin from various taxes and trade venues.

But, despite this effective and glorious reign, Basil's one failure was his lack of foresight in carrying on the Macedonian legacy. He seems to have been so obsessed with establishing the greatness of the Empire that all other things (relationships included) were of no concern.

As stated before, following Basil's death, the succession fell to his brother Constantine VIII, who throughout his entire life had avoided leadership and remained content to simply reap the benefits brought about by being the brother of a successful Emperor - wine, women, and coin. Constantine, an old man by this point and intent on retiring, eschewed his responsibility as Emperor, and died only 3 years into his reign, having achieved essentially nothing of note. This was to set the trend for further ineptitude in the Byzantine government.

To make a long story short (although I will link a previous post that elaborates more on the matter), the Empire was ruled by a series of mostly incompetent Emperors for nearly 60 years after that. The Macedonian dynasty, who ruled for two centuries, died out with the death of the Empress Theodora in AD 1056, leaving the succession a point of contention. It is during the time that Isaac I Komnenos rose to power, nominally beginning the Komnenoi dynasty, but despite his relatively effective (though short) reign, he was forced to resign due to scheming from rivals in the Imperial court. Following Isaac's departure in 1059, further scheming and incompetence continued.

Of course, in AD 1071, Romanos IV Diogenes, who began his reign with promise, was captured at the Battle of Manzikert, sustaining serious losses in equipment and manpower. When he was returned after being shamed by the mercy of the Turkish sultan Alp Arslan, he was stripped, blinded, and exiled. This act seriously undermined the authority of the Imperial throne, and Romanos' next two successors (Michael VII and Nikephoros III Botaneiates) were, on the most part incompetent and did nothing to repair the damage. As a result of this series of blunders, the Empire suffered greatly. Not only were huge sums from the Imperial treasury lost from the hiring of a great number of mercenaries under Romanos Diogenes, but the destruction of the army left most of Asia Minor wide open to attack. With few forces to defend the cities in the Byzantine heartland, the Turks were able to supplant Byzantine rule in central of Anatolia in the span of just 10 years.

So, this is the Empire inherited by Alexios - an Empire that suffered from negligence and incompetence in all realms of government for decades. With enemies set upon all fronts, Alexios' usurpation of the throne from Botaneiates was seen as necessary for the very survival of the state, since Botaneiates did very little to solve the problem of the Turks, just as his lackadaisical predecessors Michael VII, and Constantine X Doukas.

So what did Alexios do differently?

Alexios' concerted and steadfast effort (based on his impressive experience and ability as a military leader) allowed him to personally challenge opponents in a seemingly hopeless battle against vastly superior forces. His original and cunning stratagems contrasted with the conventional and uninspired military leadership of his predecessors, and his purported regard for mercy and compassion in war and peace won over the hearts of the people. He also fought in many of the battles personally, which also won him admiration from his peers.

His open resourcefulness, such as his appeal and ability to negotiate with the Latins, a course of action long neglected by previous Emperors, in the end was manipulated mostly to the benefit of the Empire, despite significant risk from the Crusaders. Apart from a few isolated incidents, the increased contact with the West during the Komnenian era did end up benefitting the Empire in the long run, both militarily and economically.

Alexios' well-planned alliances, especially with his adoptive mother, the previous Empress Maria of Alania, his brother Isaac, several notable great families, as well as his own family's history of dedication to the well-being of the Empire (most notably, the reputation of the previous Emperor Isaac, and Alexios' father and mother, the Domestikos and Domestikissa of the Scholai), and finally, his own personal eloquence and charm set him up for success on the social front.

His necessary reforms of the currency (the introduction of the hyperpyron and the stabilization of the economy), a system long in decline, was finally corrected through Alexios' prudence and pragmatism, rather than the grandiose and impractical reforms like those of Romanos III Argyros.

In the end, Alexios was a practical and effective leader who knew well the failures of his government and sought to correct them through his own ability and desire for the well-being of the Empire. He did not limit himself to old traditions and systems, but sought his own solutions that in the end brought the Empire back from the brink.

Besides the thoroughly awesome account provided by Ambarenya, I must also point out that Alexios I Komnenos also altered the relationship between Byzantine aristocratic families and the Imperial government.

According the Michael Angold in his book, The Byzantine Empire 1025-1204, a Political History: "The aristocratic principle triumphed with Alexius Comnenus. He did everything to root the aristocracy in the foundation of the state and thus transform it. His reform of the honours system meant that their position at the apex of society was enshrined in the court hierarchy" (page 156) and that "The character of the regime brought into being by Alexius Comnenus's coup was becoming clear. Power was now in the hands of an alliance of aristocratic families, all united by close marriage ties" (page 128).

The honours system that is being referred to was a series rankings and titles. Alexios apparently got rid of these and created a new set based on the term Sebastos. These titles were given out only to family members or those families the Komnenids were married to (page 128).

By incorporating aristocratic families into the Byzantine system of government, Alexios could rely on their support and at the same time those in command could be trusted. This would prevent a repeat of situations like the Battle of Manzikert, where the defeat of Romanos was ensured by the betrayal of Andronikos Dukas, who retreated with the intact rear-guard, meaning that the portion of the Byzantine army that was with Romanos had no means of refuge (Page 45).

Noble families who wished to gain political power now had an incentive to work with reigning emperor, rather than rebel or betray him to ensure their particular "faction" could gain power. Now multiple factions could cooperate rather than compete through the use of marriage alliance and their subsequent incorporation into the ruling class.

Prior to this the dominant "party" would have the emperor and his immediate family, and that other families would have been excluded from direct power since the emperor would have relied on specially selected servants and individuals. Power was essentially a zero-sum game. Either your family was dominant, or it wasn't. Alexios allowed a network of families to come to dominance, meaning that there were multiple stakeholders, and the emperor would not have to take risks giving an aristocrat military command.