First, I might suggest to you that learning to play the guitar (which is really easy) can provide the background chords to accompany you while you sing. I taught myself how to play chords on the guitar for that specific purpose years ago. But, if you want to record yourself on your computer, here is what I use to achieve the kind of broadcast quality sound that you need if you ever want to sell your music over the Internet or get it played on the radio.
Microphone - http://www.guitarcenter.com/Rode-Microphones-NT1-A-Condenser-Mic-Bundle-105619399-i1474748.gc
Microphone stand - http://www.guitarcenter.com/On-Stage-Stands-Heavy-Duty-Tele-Boom-Mic-Stand-100235792-i1528172.gc
XLR microphone cable - http://www.guitarcenter.com/Monster-Cable-S-100-XLR-Microphone-Cable-102469608-i1128553.gc
Computer recording interface with built in broadcast quality sound card and recording software - http://www.guitarcenter.com/Avid-Pro-Tools-LE-Mbox-Mini-3--3rd-Gen--106091184-i1552697.gc
Studio Quality Headphones - http://www.guitarcenter.com/Sennheiser-HD428-Closed-Back-Around-the-Ear-Headphone-106498116-i1567482.gc
Those are the essential thing you need to record yourself singing. The Pro Tools recording software that comes with the Mbox has rhythm track software included in it.
ADDITIONAL things that will enhance your capabilities and make your recordings sound more professional include:
BOSS VE-20 Vocal Performer - http://www.guitarcenter.com/Boss-VE-20-Vocal-Effects-Processor-105654529-i1474806.gc?esid=BOSS+VE-20
TuneTracks rhythm software - http://www.guitarcenter.com/Toontrack-Superior-Drummer-2-0-105153522-i1401765.gc
Yamaha keyboard (plays background music and has lots of rhythm tracks) - http://www.guitarcenter.com/Yamaha-PSR-E333-61-Key-Mid-Level-Portable-Keyboard-106701758-i1881151.gc
Roland SP-555 Sequencer and beat producer - http://www.youtube.com/watch?v=7HTQ6Nm6Z9c
Magix Music Maker (create your own music) - http://www.magix.com/us/music-maker/premium/?AffiliateID=29&partnerid=9031&campaign=Affiliate%2FGoogleAffiliateUS