What license should I get to make the recording legal?

You will need to get a license from the entity that is responsible for licensing the professional artist's version of the song as well as the underlying song. The professional artist may not be willing to allow you to use his work without specific permission as this is well beyond a cover of the song.

In theory, you would need both a mechanical license and the masters use license. However, you say it is you "singing alongside with an already existing song performed by a famous artist." Do you mean you are singing along with Justin Timberlake, for example? Unless you have created some sort of great remix, Timberlake's record label is not going to give you a masters use license for this purpose. However, if you mean it is just you singing along to a karaoke version of a famous song, then yes, you need a mechanical license, which is easy and inexpensive to get, and a masters use license, which is not easy to get and may be expensive. However, if all you are doing is posting this to youtube, in most instances, the worst that will happen is that the video will be removed. In most cases, the video will not be removed and you will be able to show off your singing skills to your friends and family. Theoretically, you could be sued for copyright infringement, but that is unlikely. Most music artists consider such sing-along videos to be a compliment from a fan, and they are happy to see you show your appreciation in this way. If all you have is the audio and you want to put it on youtube, you can lay down any picture as the visual component of the video. Youtube now has a built-in editor, so you can create the video right there on youtube without owning any software. However, if you plan to sell the recordings, rather than just putting the song on youtube for fun, you need both licenses. If you use an aggregator service to get onto Itunes or another service, you will need to show that you have proper licensing. If you plan to get a masters use license, contact a lawyer to help you. If you need a mechanical license, you can get that easily using song clearance services through Limelight or Harry Fox. These mechanical licenses are for songs recorded, produced, and sold in the U.S. only.