It is possible, but seriously not advisable, to split the cable. To do so, wire each end to a Cat5 Jack (or plug) using the following pinouts (cut short pieces of the inner conductors for the unused pins, so everything will line up correctly if using plugs) Org/Wht J1-P1, Org J1-P2, Grn/Wht J1-P3, Grn J1-P6, Blu/Wht J2-P1, Blu J2-P2, Brn/Wht J2-P3, Brn J2-P6. This probably won't meet the specs for crosstalk, and don't even try running Gigabit on it, but will generally work for 10/100 connections. It's ugly and non-standard, but hey, so am I, and I still work...
Daisy chaining connections, like analog phones, won't work (at least not well).
The switch solution is simpler and better. Set the switch near computer B. Connect the existing wire from the Linksys to one port of the switch, and computers B and C to the other ports with standard Cat5 cables.
The third option is to use a wireless card or USB solution for Computer C, since it seems you have a wireless router already.