As Marty and Bobbi Culp (above) have proven, it's possible to do cover versions of popular songs in a way that best suits your own musical style, and also have a lot of fun doing them.
The title of this post may be "Doing a cover the right way...", but there isn't any right or wrong when it comes to music - as long as you're doing it, that's the only important thing. When we're young, it can be instructive to emulate our heroes; singing their songs and playing their riffs over and over is a great way to get fluid on an instrument, and gain vocal technique.
It's when we stop trying to sound like someone else, that's when our own individual style automatically emerges - it's basic human nature, and it's a good thing. Some people never stop trying to emulate others, and some players never even tried to; probably, all of the most creative and truly unique musicians and singers in musical history fall into the latter category.
Creating your own art is the most satisfying experience in life, period. But, it's fun to do a cover every once in a while, and whether you do it exactly like the original version, or if you put your own personal creative spin on it, it's all the same, it's all good. That is, as long as you don't get too obsessive about any aspect of it. At the risk of sounding pedantic, or being a sensei, I'll just throw this out: trust your own instincts, believe in the song and believe in yourself, and remember to give back to the music, something from your own soul, as much as you've received from it.
In no particular order, here are a few examples of great cover versions; some are closer to the original recordings, and some are so far out in the cosmos that they're still orbiting some distant musical sun.
First, The Beatles performing the Isley Brothers' "Twist And Shout":
Here's the Grateful Dead, jamming on Buddy Holly's "Not Fade Away":
Jimi Hendix's "Purple Haze", as rocketed into the galaxy by the Kronos Quartet:
Matthew Sweet and Susanna Hoffs singing Neil Young's "Cinnamon Girl":
"Walk Away Renee", originally done by the Left Banke, as re-imagined by Linda Ronstadt and Ann Savoy:
Wilson Pickett, with (I never knew this before) Duane Allman on guitar, doing his rendition of The Beatles' "Hey Jude":
James Taylor's bittersweet social awareness ballad, "Millworker", totally slammed in a powerful performance by Bruce Springsteen:
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Part 1 of this series, "It's Only Rock And Roll, But Maybe Not".
And Part 2, "How I Joined A Cover Band, And Survived To Tell About It".