Help me see if I understand this RLC circuit
The orthodox way would be via transfer function/Laplace transformation.  Just a few hints:

U_c(s) = E(s)*Z(s) / (Z(s) + R_2). 

E(s) is the transformed step function, Z is the parallel complex resistance of C and R2, s is the complex frequency.

The step function's Laplace Transformation is 1/s.

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(Some easy algebraic calculation in the frequency domain)

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The result if re-transformed (look up tables, f.e. Bronstein-Semendjajew) into the time domain will be the same like the vC(t) mentioned at the end in the question.

The current can be calculated in the frequency domain by dividing U_c(s) by the complex resistance of C. Then another re-transformation/look up is needed.

Those familiar with the Tellegen theorem could simplify the algebraic calculation.