There are several ways to do this

-electroplate copper (typically requires that the tube be made slightly conductive (typically by using a graphite coating or similar).  You can find more info. on plastic plating from finishing.com or by searching for the term electroplating plastic on Thomas Register or Google.
-use a thermal reduction from a copper sulfate(?) (maybe it was chloride) solution (requires that the tube wall be "wettable" by aqueous solution).  The idea is to wet the surface, then dry and apply heat sufficient to decompose the salt, leaving metal behind.  Works well on glass, but the temperature required for reduction will probably melt plastic.
- coat the i.d. with a conductive paint (see Acheson Colloids Company)
- vacuum metallizing or ion deposition (probably won't be able to coat very long tube)

Can you not just thread a copper wire down the tube, and stake the ends (using epoxy or similar) to hold the wire in place?

You can buy copper tubing coated on its O.D. with plastic (LDPE I think).  Could you use this as plastic tubing with copper on the I.D.?