String gauge is the important thing for this, especially if you're going to be on a 4 string neck with a 34" inch scale. If you look at the following link http://circlekstrings.com/CKSIMAGES/CircleKtensionChart.pdf you can calculate the tension you currently have your strings at with your current tension, and what you would need to increase the gauge to to get an equivalent tension in C. For example, if you're tuned to E on your low string with a 0.105 gauge string, there will be approximately 43" pounds of tension. To get the equivalent tension in C, you would need a 0.130 to 0.135 gauge string.
I have my bass tuned in drop C, and I was using the low 4 strings in a 5 string set (130, 100, 80, 60), but I found the tension in the high C and F strings quite high, so I'm currently getting the bass set up with a 0.125 set, but using the low B string for the C, ignoring the E string, and using the other strings in their respective places (0.125, 0.85, 0.65, 0.45). This should hopefully lead to a balanced tension set, with the string tension roughly the same across the strings