What modern celebrity could be compared to Achilles of The Iliad?
As I read your question, I started thinking about people the world thinks of as "having it all" but who somehow "blew it" because of one fatal weakness. You use the word "celebrity," so I assume this needs to be someone recognizable as well as someone who is somehow seen as a hero. Two men came quickly to mind: President Bill Clinton and Tiger Woods.
President Clinton was arguably the most powerful man in the world when he was in office. He had a powerful and supportive wife, a loving daughter, and he was in the enviable position of being able to call any ruler of any country in the world and at least be heard. He had everything, including the love of the majority of the American people. And then came the Monica Lewinsky scandal and his eventual impeachment and censure. What a low moment for such a powerful man; he succumbed to his one failing--women. Clinton, of course, recovered from this personal failing (actually many failings, since this was just the most public incident of many) and is still beloved and sought-after by people all over the world.
The second celebrity who had a similar failing, though it looks different, is Tiger Woods. Arguably the best golfer who ever lived, Tiger just seemed to have it all: money, fame, athletic ability, a lovely wife and kids, you name it. Then it all blew up. His numerous extra-marital affairs became public and he lost it all, including many of his fans and nearly all of his sponsors--and even his stunning ability to play the game at a level no one ever has. His inability to be faithful to one woman was his achilles heel, and it remains to be seen whether he, like Clinton, will recover.
One of the previous answers mentioned that Achilles is a character of great physical strength and someone that perseveres through whatever difficult situation is thrown at him. These are heroic qualities about Achilles. Additionally, Achilles's sense of honor and excellence is a major part of his character. He simply wants to be the best that he can be. This kind of focus also makes him an intense individual. He's also quite a violent and brutal warrior.
With the previous characteristics in mind, I feel that Dwayne "The Rock" Johnson is a good pick for a modern celebrity that compares to Achilles. I'd like to clarify that much of Johnson's similarities are because of the characters that Johnson has played in his film and wrestling career. It's hard to separate him from those characters. That's how he is seen. He's big and muscular. He oozes a warrior mystique about him, and he's brutally violent in a number of films. Additionally, movies that have him playing a hyper-masculine warrior person tend to have a scene with Johnson intensely staring down his opponent. Outside of his characters, people that work with him describe him as being intense and focused.
There's an intensity when you're working around him and you can see it and feel it in the air. He's going 150 miles an hour every day.
Johnson simply desires to be the best that he can be. He steps wholeheartedly into each task that is before him. It's probably why he is able to pull off such diverse rolls. He's not only an action star, but he's also been in several successful comedies and kids' movies.
First, figure out the characteristics of Achillies. What makes him a hero? Think of what a hero is in general. Are you going to approach in terms of physical strength or simple perseverance. If you approach your essay from the angle of a hero (Achillies) as someone who never gives up no matter how dangerous the consequences, then you have a lot of famous people to draw comparisons with. Anyone who has fought for equality would be a great example (Susan B. Anthony, Malcolm X, Elie Wiesel). You could go political and think of Presidents who persevered (Washington, Lincoln, FDR...). I suggest making a list of Achillies heroic characteristics, and think if them in a broad sense. Once you have this list, begin to brainstorm famous people who you know that have gone through similar struggles. If you need to stick with celebrities, Robert Downey Jr. has many struggles! I hope this helped. Once you have your characteristics listed, and if you can still not think of anyone, let me know.