Deriving Confidences from Distribution of Class Probabilities for a Prediction
If I have Classes [C1, C2, C3, C4, C5] and my Probabilities are {C1:
  50, C2: 12, C3: 13, C4: 12, C5:13} my confidence in predicting C1
  should be higher than if I had Probabilities {C1: 50, C2: 45, C3: 2,
  C4: 1, C5: 2}.


Assuming that those probabilities are accurate, this isn't true. In your second case you can be a lot more confident that the ground truth is one of C1 or C2, but in terms of absolute confidence about C1 the probability is the same across both examples. To illustrate this with a more clear example, if you had a 100 sided die that had 50 sides labeled with "C1" then the labels on the the other 50 sides are irrelevant to the likelihood that you would roll a "C1".

Now with that said, your probabilities from your model are most certainly not perfect so there may be a way to use the intra-class correlations to improve them. Can you provide some more details about your specific problem and modeling workflow that you have used to get your probabilities?