About 4 years ago Chelsea was an upper-mid table London club that were on the brink of bankruptcy. A Russian oligarch Roman Abramovich bought over the club and spent a lot (like a LOT LOT) of cash on signing new players (and paying their high salaries), upgrading their training ground and academy etc.
So Chelsea's won the Premiership (league) twice in a row (going for third this season) and generally been the team to beat. Fans of other teams are obviously unhappy, especially the ones that used to be the best teams around, ie Manchester United and Arsenal.
Chelsea has built success by taking spending on signing some of the best players in the world to new extremes and supporters of other teams resent that. The biggest clubs have been signing good players to maintain success and their position over other teams for many years, but not to Chelsea's extreme so people think that this time it's different from back then.
On the field, Chelsea has built a reputation of consistancy. Even when they're not at their best they can grind out results and that's what seperates them from their competitors. They may not win pretty but they win almost all the time.
An issue people have is that with all the good players Chelsea should be palying better football than they usually do. Another thing is players that would easily make the first team of any other club are spending a lot of time on the substitute's bench.
Lately Chelsea has been viewed as a kind of young talent predator, signing good young players from around the world from their original clubs. Lately Leeds has reported Chelsea for illegally (according to Leeds) tempting some of their youngsters to London.
In summary, Chelsea's ruffled many feathers on their way up. The people that complain the most are the ex-winners that have found it very difficult to keep up with Chelsea so far.