Find a user-friendly website for beat makers. Listen to some of their samples to see if they have what you're looking for.
Start off with a basic drum sample (you can change it later) or begin with treble clef sound samples like guitar, keys, horns, wind instruments or synthesizers. Playing with one sample or sound at a time will focus your track as you progress through the bars, and you can embellish it more later on. Tweak the bass and drums last.
With the basic structure of your beat programmed, take a moment to consider popular song form. The verses of most songs you hear on the radio hang in the 16 to 32 bar zone, with the chorus usually being closer to the 16 bar range. Be creative yet consistent with bridges, vamps and the pre-chorus.
Study the greats. Take a closer listen to the music that first inspired you to grab a mic and spit your own rhymes. Hear the quality and depth, or sparseness, of the tracks used by the artists and producers you'd like to collaborate with, and make sure the flow of your beats can hold water next to theirs.
Once you've put your Grammy contender's best into your first few beats, most free sites will require you to sign up for a paid membership in order to get your beats in MP3 format and save them offline. Some sites, like JamStudio, allow their users to email a link to their tracks to friends for feedback before committing to a membership. Being able to allow others to hear your tracks from their own computer is also great if you are collaborating with other musicians, planning to use the beat on a mixtape, or if you hope to feature other rappers on the track.