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Here is the process for altering the colors of a circle multiple times with a timer in a Java graphical user interface:

1. Create a new Java graphical user interface (GUI) project in your IDE.
2. Add a JPanel to the JFrame, which will be used as the canvas for drawing the circle.
3. Create a Circle class that extends JPanel and implements ActionListener. This class will contain the logic for drawing and updating the circle with different colors.
4. In the Circle class, declare a timer object and a counter variable that will be used to keep track of the number of times the circle's color has been changed.
5. In the constructor of the Circle class, initialize the timer object and set the delay to a specific time interval (e.g. 1000ms).
6. Override the actionPerformed method of the ActionListener interface, which will be called each time the timer fires.
7. Inside the actionPerformed method, update the color of the circle with a new random color using the Java Color class. Increment the counter variable each time the color is changed.
8. Call the repaint method of the JPanel to redraw the circle with the new color.
9. In the main method of the GUI project, create an instance of the Circle class and add it to the JPanel.
10. Run the project and observe how the timer changes the color of the circle multiple times.