This is an extremely good point. My old understanding was that it could like for 4 days outside the body, however it has been shown the virus can remain viable for 6 weeks
http://jid.oxfordjournals.org/content/early/2013/11/22/infdis.jit648.abstract
There is further discussion about this here:
https://blog.aids.gov/2014/01/scientists-discover-hepatitis-c-virus-can-remain-infectious-outside-of-the-body-for-up-to-6-weeks.html
I have personally seen the parents of an HCV son acquire acute Hep C due to razor sharing so there is no doubt the virus can survive and infect from this source.
Viral loads fall rapidly on treatment so after a couple of weeks there will be very little (or no) virus in your blood to coat either a razor of a toothbrush, so if you are contemplating frequent replacement of razors and toothbrushes there would be more benefit in doing this daily at the beginning of treatment than at the end.
Here is what the CDC has to say about disinfection:
http://www.cdc.gov/hicpac/Disinfection_Sterilization/3_2contaminatedDevices.html
So soaking razors and toothbrushes in a 1:10 dilution (1 part bleach, 9 parts water) would be an acceptable alternative to replacement.