In order to be a successful lyricist, you must be a creative person with the ability to put your thoughts down on paper in an artistic manner. Often, songs are full of different types of meter, rhyme, or imagery. Many songs are poetic pieces with a musical overlay. If you are able to write poetry, there is a good chance you would be able to write lyrics.
The key to being a writer in any field is simply to write often. You should always be working on your writing skills by keeping a journal, or having ongoing poetical works, stories, articles, or essays. It is not good to let your cognitive creativity go into atrophy, as it will affect your success as a lyric writer.
It would be a helpful first step for a beginner lyricist to find instrumentalists to whom they can submit their lyrics to, and see how they are able to be meshed together into a song. This would be excellent practice, and it would also help to have connections with people who may be able to help further your song writing career.
Another advantage for a promising lyricist would be to take a few musical theory classes and learn the basics of a common instrument such as the piano or guitar. The main focus of a lyric writer's career is obviously the words in a song, but being able to strum a few chords and intelligently discuss the musical progression with the composers you work with, is definitely a bonus.