Below is verbatim your system. Copy it and paste it into a cell and run it.

eqns = {-b[t]*kr + vc[t] + d2 - b'[t]*kl == 0, bc[t] + c1*vc'[t] == 0,
   vl[t] - kr*b[t] + kl*b'[t] == 0, vc[t] - d2 - vl[t] == 0, 
  vin - q1*2*b[t] - d1 - vl[t] - q2*b[t] == 0, b[t] + bc[t] == 0}

Next copy and paste the following instruction into the next cell and run it as before.

Reduce[eqns, {b[t], vc[t], vc'[t], b'[t]}]

The output allows you to solve your problem by considering 5 cases. In each case, determine b and vc and substitute them back into eqns to find out whether there are additional constraints.

I hope that the above suggestions will help.

Your system is indeed overdetermined. One thing you can do is to solve the first four equations and then replace into the remaining two, to get the compatibility conditions:

sol = DSolve[eqns[[{1, 2, 3, 4}]], {b, vc, bc, vl}, t] // First
compatibility = Solve[eqns[[{5, 6}]] /. sol]