RC step-response and how to go about voltage across capacitor
You are going about this in a complicated manner when, in fact you can convert the three resistors into one resistor and, the voltage source becomes 3 volts. So, the 6 volt voltage source, R1 and R2 form a potential divider that has an open circuit voltage of 3 volts. 



Looking into that circuit you should be able to see that the equivalent resistance is just R1||R2 or 5000 ohms.

So now you have a new circuit that is a 3 volt source in series with 5000 ohms connected to R3 in series with C. You can simplify further to put R3 (also 5000 ohms) in series with the previously calculated 5000 ohms from the parallel connection of R1 and R2.

The final circuit you end up with is 3 volts feeding C via 10 kohms.

Can you take it from here?