I use cross validation to find a best set of parameters for random forest on my dataset. Then I use the best model to fit my train set and got an average AUC of 0.6883. But I can see the variability of the AUC scores was significant, from 0.5113 to 0.8068. Can I claimed my model performance was 0.8068 (as what it achieved best) and use the corresponding model estimator to fit the test set for prediction?
As mentioned in the comments, you cannot use your training set to claim anything about your model's performance. You need to use a cross-validation scheme there as well. This is called nested cross validation.
See the following great question and answers for more details: Nested cross validation for model selection