Is it legal to have online tournaments and charge players an entry fee?

The main hurdle I see is that the game company owns the trademarks on the name, characters, and images, as well as copyright on characters, images, and other elements. If you want to run a tournament legally, you need to be working with the game company on it. At that point, your lawyer and their lawyer will figure out the particulars of how it can be done. This is why most game tournaments are done locally, quietly, and among friends only. Whenever you try to make money off the use of someone else's intellectual property, you are going to run into complications. However, do not let this stop you from contacting the game company and trying to make a go of it. They may warmly welcome a partnership with you. If you have a website and the means to run an online tournament and have the ability to gather players, many game companies would consider you quite an asset.

The short answer is Yes. You can run skills based competitions with entry fees and a percentage admin fee. This is because in about 38-40 states there are exceptions to gambling prohibitions allowing the competitors themselves to wager on the results of their own efforts. The process must be carefully established to make sure the resulting competition fits within these exceptions. Make sure to engage an Attorney with such expertise. This is not a venture you should explore with out expert help.