You can try using anomaly detection.
Make your 'features' number of likes/spam/friends/other relevant features you've found helpful, and use the algorithm to detect the anomalies.
Another approach could be supervised learning - but will require a labeled set of examples of "fake" and "real" users. The 'features' will be similar to these for anomaly detection.
Train your learning algorithm using the labeled set (usually referred as training set), and use the resulting classifier to decide if a new user is fake or not.
Some algorithms you can use are SVM, C4.5, KNN, Naive Bayes.
You can evaluate results for both methods using cross-validation (this requires a training set, of course)
If you want to learn more about machine learning approaches, I recommend taking the webcourse at coursera.