Replacing splined cane
First, remove the old cane. Start by using a utility knife, chisel or something to break the bond between the top of the groove and the spline (if you don't, you might splinter away a bit of the seat. No big deal on the inner edge, but will be seen on the outer edge when done). Pry the spline out with an appropriate tool. I use a 1/8" mortice chisel, but you can get a spline removal tool from the cane suppliers. Once the old spline is out, clean out the old cane, glue etc. from the groove.
To replace the cane, cut the cane mat oversize and soak it a while in warm water. Lay it over the seat and wedge one edge (I start on the long sides) into the groove. (I use a bunch of pine wedges I made up). Make sure your cane mat is square to the seat frame as you go. Once the first side is wedged into place, wedge the opposite side, then do the short edges.
Using a sharp chisel, trim the cane below the top of the outside edge of the groove.
Now insert the spline. Your seat may have been splined one of two ways. Some seats are splined with four pieces of spline, mitered in the corners. In this case cut the spline to length, cut a 45 degree angle on the ends, and insert them one edge at a time. Other seats a splined with a continuous spline. On these seats, start at the middle of the back long edge, work your way around the seat, and when you get back to the starting point trim the spline to length to butt against the beginning. Note that the size spline you use is matched to the width of the groove. The suppliers' catalogs will indicate this.
When you are ready to spline, Remove enough of the wedges to get you started, squeeze some Titebond III (or similar) glue into the groove, lay the spline in, and hammer it in using a wooden block as a "cushion". Work your way around the seat, and when all the spline is in, clean up any glue that has oozed out. Don't worry if the seat seems a little slack - it will tighten up as it dries. Once the glue has dried, you can varnish the seat and cane if you desire.
That's all there is to it. The first time might take a while as you work it out, but after that a seat can be done in under an hour or so.
Have fun,
Dan