For me, there are two aspects to a great league: quality and competition.
Nobody cares about a league with just one team dominating, because it would be no fun to watch, even if that's the best team in the world. Similarly, though, nobody cares about a league with lots of teams having a chance if they are all dreadful and insignificant on the world stage. Competition and quality. You need a balance between the two.
To analyse the Premier League's claim to be the best, then, I'm going to compare it to two rivals for the crown (Spain's La Liga and Germany's Bundesliga) along these two lines. This won't be a comprehensive analysis, and there are other relevant factors and contenders, but it should at least establish a reason to take that claim seriously.
In my mind, the English league is far ahead when it comes to competition. Despite fears of a "big four" (Manchester United, Chelsea, Arsenal, Liverpool) dominating the league, taking the top four spots in five out of six seasons from 03/04 to 08/09, recent developments have broken this up nicely. Liverpool have spent a large time out of contention, overtaken by teams such as Manchester City (who have since won the title twice), Tottenham Hotspur (who finished fourth in 09/10), and local rivals Everton.
Chelsea also slipped out in 11/12, and United did the same in 13/14. That only leaves Arsenal as a consistent performer, and they have often only secured fourth place with their final game of the season. Even if the "big four" had remained, however, the league would still be far from boring. Four times competing for the title makes an exciting enough contest, especially if they are evenly matched, and especially if their positions within the top four spaces vary year-on-year.
It's certainly a lot better than other leagues, which seem to have a "big three" or "big two" system. In the last 10 La Liga seasons, for instance, Barcelona and Real Madrid have occupied the top two spots on all but two occasions, making the league a little stagnant. Last year's victory for Atlético Madrid ended a 9-year streak of Barca and Real champions, and even then the two teams finished a very close (3 points off) second and third.
In Germany, Borussia Dortmund and Bayern Munich have similarly monopolised the top two spots for the past three seasons. Bayern finished the last two years 26 points clear of third place, with Dortmund 17 points clear the year before. In the three cases before last year's upset, those numbers were 24, 39, and 25 in Spain. In England, the distance between 1st and 3rd has been 4, 14, 19, 9, 11, 7, 4, and so on, suggesting that the race for Premier League top spots is much more open and competitive, and frequently goes to the wire.
This is also reflected in the distance between the top two, which in England has been 2, 11, and 0 in the past three years. For Spain, it has been a respectable 3, 15, and 9 before Atlético's upset, suggesting that whilst Real and Barca are distant from the pack they can at least keep each other in check. In Germany, the gulf has been 19, 25, and 8 points, which means that the season is often over long before the end. As Bayern now have a habit of buying Dortmund's best players, such as World Cup decider Mario Gotze and Bundesliga top scorer Robert Lewandowski in 2013, the gap between these two rivals seems likely only to widen.
As if the German trend for single-club dominance needed to be highlighted further, in each of those past three seasons the Bundesliga champion has also won the domestic cup, a "Double" which is seen as rare in England (only once in the last ten years) and Spain (twice in the past ten). Whilst La Liga might well be said to have a dominating "big two", therefore, the Bundesliga could be described as featuring a "big one". England's "big four" looks dynamic in comparison, especially with the fall of United and rise of City. At respective points during last season I wouldn't have been surprised to be told that any of five teams could have won it.
Somewhat ironically, the Premier League leaves its rivals well behind when it comes to competition, although the arrival of Atlético may rejuvenate the Spanish league into a more respectable "big three". It remains to be seen, however, whether English clubs can hold a candle to their continental contemporaries when it comes to quality.
Champions League finals are a good place to start. Followers of the European competition in 13/14 will have noted a pattern of Spanish dominance, culminating in an all-Madrid final, and the previous year saw something similar happen with Bayern securing their treble against Dortmund. In 11/12, though, Chelsea brought the trophy to England, and in 10/11 Manchester United also reached the final. In the past 10 seasons, we've had four Spanish champions (Barcelona 3, Real 1), three English (Liverpool 1, United 1, Chelsea 1), two Italian (Internazionale 1, Milan 1), and one German (Bayern 1).
More English clubs have won the Champions League in that period than clubs from any other country, demonstrating not only the quality in the Premier League but the way it is shared around rather than concentrated. When losing finalists are taken into account, four English clubs appear 8 times out of 20, compared to three Spanish clubs appearing 5 times, two German clubs appearing 4 times, and two Italian clubs appearing 3 times.
English teams have constantly held their own against the continent's elite, with the Premier League also boasting more strength in depth by varying its tributes. This is reflected in the coefficients assigned by UEFA, European football's governing body, which place the English league second for strength between Spain and Germany. The Bundesliga hasn't had a team rated top in Europe (over a 5-year period) since Bayern from 1986-90, but Chelsea (2004-08) and United (2007-11) have both featured recently. With that said, there is no denying that Barcelona and Real Madrid have held the title for by far the longest.
Moving on from UEFA rankings and continental performances, we can look at what the players and clubs themselves think by observing their transfer histories. Much has been made of the fact that Gareth Bale and Luis Suarez, winners of the PFA Player's Player of the Year for the past two seasons, immediately moved to Spain after accepting the award. This implies that La Liga is seen as a step up for the Premier League's top player, like a top Championship striker being bought by a team in the top division.
However, I feel that an exception needs to be made for these two teams. Unquestioned domestic dominance means that Barcelona and Madrid can comfortably focus more on their brand than sensibly securing their place in the league, a luxury that competitive English teams do not enjoy. A Premier League team would be unlikely to splash £85 million on Bale, or £75 million on Suarez, and as clubs Tottenham and Liverpool were hardly dominant they would have been foolish not to accept such a massive amount of money to spread around the squad.
It is also important to note that this traffic has been anything but one-way. Beyond the unsellable Lionel Messi and Christiano Ronaldo, the highest goal tallies in La Liga in those two seasons came from Radamel Falcao (28), Diego Costa (27), Álvaro Negredo (25), Roberto Soldado (24), and Alexis Sanchez (17). The five highest assist tallies were recorded by Ángel Di Maria (17), Andrés Iniesta (16), Mesut Ozil (13), Cesc Fabregas (13), and Koke (13). Eight of those ten players are now employed by Chelsea, City, Tottenham, Arsenal, or United, whereas English exports to Spain invariably go to the two dominant teams. Just Iniesta and Koke remain.
To contrast, Robin van Persie, Wayne Rooney, Daniel Sturridge, Eden Hazard, Sergio Aguero and Yaya Toure have all stayed put, so it isn't exactly true that the Premier League can't keep its best players. If anything, there have been more La Liga arrivals than departures, including some of the divisions' very best. For example, Chelsea kick off the 14/15 season having finished third last time around, and yet they have managed to sign three players from the Spanish champions (Atlético's top goalscorer Costa, starting left-back Felipe Luís, and La Liga's best goalkeeper Thibaut Courtois) and one from the runners-up (Barcelona's Fabregas).
Spain might edge England for quality, but it's hardly an open-and-shut case, and they lose their competitive balance in the process. Looking at these two qualities, therefore, we can see that the Premier League may be far from perfect, but that it certainly has enough to compete with its rivals. Even if it's not the best domestic championship out there, then, I'd say that the most-watched league in the world is certainly in the running.