Q:

How to get a next alphabet in alphabetical sequence in apex class? 

How can I get the next alphabetical character in alphabetical sequence? 
ex.  I have  'A' letter, I want 'B' by using 'A'. 

A:

The methods you'll want to use are listed in the Apex Developer Guide under  String Class, specifically charAt and fromCharArray. 
Integer charCode = 'A'.charAt(0);
List<Integer> chars = new List<Integer>{charCode+1};
String nextChar = String.fromCharArray(chars);
System.assertEquals('B', nextChar);

A:

You can use String.indexOfChar to find the next value in a simple string.  Here's an example:
public static String getNextChar(String oldChar) {
    String key = 'ABCDEFGHIJKLMNOPQRSTUVWXYZAabcdefghijklmnopqrstuvwxyza';
    Integer index = key.indexOfChar(oldChar.charAt(0));
    return index == -1?  null: key.substring(index+1, index+2);
}

This includes automatic wraparound from Z -> A and z -> a.

