Intarsia is circular knitting's weak point. The easiest thing to do if doing intarsia, is to knit it flat, and seam it.
It is possible to do intarsia "in the round," but it's not truly in the round. It's sort of a combination between working in the round and working flat. You basically work a round, and turn the work and purl back, but you do it in a way that joins the piece as you go. You will have a sort of seam at this turning point.
I described the technique in an old thread, which I just dug up. I'll cut and paste the instructions here, in case you're interested....
The idea is, you knit your way through one round, and then purl your way back (unusual for circular knitting!). You choose the point where you will be turning the work, a place where a seam-line will be least noticed. You will increase an extra stitch here before you begin the first round of intarsia. This stitch will be the "seam", and will be slightly different looking than the other stitches. So, you create this seam stitch, and knit all the way around, doing intarsia, ending by knitting this seam stitch. Turn the work, and slip the seam stitch. Purl your way back around to the seam stitch and purl the seam stitch. Turn the work. Slip seam stitch, and work the round, ending by knitting seam stitch. Continue this way, always beginning a round by slipping the seam stitch, and ending the round by knitting or purling it.
KnittingHelp Queen Bee
"There is a vitality, a life force, an energy, a quickening that is translated through you into action, and because there is only one of you in all of time, this expression is unique. And if you block it, it will never exist through any other medium and it will be lost. The world will not have it. It is not your business to determine how good it is nor how valuable nor how it compares with other expressions. It is your business to keep it yours clearly and directly, to keep the channel open. You do not even have to believe in yourself or your work. You have to keep yourself open and aware to the urges that motivate you. Keep the channel open."
— Martha Graham