Would it be possible to make a game based on a series, even though the series is copyrighted?

Anime rights are a mess because you're dealing with multiple companies in multiple countries. If a show has been licensed in the US, such as DeathNote, then it would be the responsibility of that US company to enforce its copyright on the characters & story. However, the actual character art and all that is owned by the Japanese anime company. Even the songs and music may be owned by several different companies in Japan, a fact that has screwed up the hopes of many anime from ever coming to the US in any sort of legal way.

If you were just doing this for yourself and not releasing it to the public, it would be fine.

If you were going to release this to the public, but not for money, it's kind of a grey area. Yes, you are violating copyright, but you are also allowed to use copyrighted material on a limited basis, such as for a review, or satire. This called "Fair use" but it's kind of hard to define sometimes when you've gone beyond "fair use" and into "copyright infringement."

If this series had never been officially released in the US, then there wouldn't be a problem. Traditionally, Japanese firms have largely turned a blind eye to what happens outside of Japan. They figure (incorrectly) that since they aren't selling in those markets, they aren't losing any money. Furthermore, they figure such activities just serve as free advertising.

At the same time, a trip to Artist's Alley of any anime convention, and there's copyright infringement galore going on there, considering many artists are selling works they created, based on various anime/manga characters - many of which are licensed in the US. However, it's up to the (American) company to enforce their copyrights.

Some companies do - like Nintendo America who recently cracked down, hard, on a fan versions of Pokemon and Metroid games. Other companies...not so much.

Legally, no.
SOmebody else owns Death Note, and everything to do with Death Note.
Unless you have direct and explicit permission, which would require paying a lot money for a licence, you can not make anything associated with or using any content from Death note (or any other copyright material).
No matter what other material you make yourself for the game, you can npt use material which somebody else owns through copyright.

I think if you were to make the game yourself without permission that certain things would have to be changed so you don't infringe on any copyrights. If you so happen to make a really popular Death Note game and Tsugumi Ohba somehow found out you could face a suit. I would change a few key names and places around. Anything you think might be copy written. Also, if you actually do it, let me know. I'm a huge DN fan. :)