I had to do something similar in a previous project. If you want all the calculations done from the client, you'll have to implement the algorithms for Equal Interval, Equal Area and Natural Breaks yourself, there is alot of pseudocode online to do this.
But, If you implement the algorithms yourself, you probably won't be able to simulate the exact classifications you'd get from ArcGIS Desktop.
Maybe if you made a webservice with a "Classify" function which takes the (doubleArrayValues, longArrayFrequencies, ref NumClasses) parameters, you could then have the server use the ArcGIS Desktop API and then via the webservice send back the breaks to the client layer's ClassBreaksRender.