This is a situation in which you need to tread very carefully. Many companies frown (for obvious reasons) on previous employees contacting their clients. However, if you have had a good relationship with the client and are giving notice to your employer, it might be a good idea to e-mail the client about the transition as well so they aren’t blindsided.
Hopefully, you are leaving the company on good terms. Given sufficient notice, your supervisor may even suggest sending an e-mail introducing the person taking over for you.
Here is my suggestion for a goodbye e-mail to your clients. Again, be sure to use your discretion and check with your supervisor.
I wanted to email you to let you know that as of (DATE) I will no longer be working for (COMPANY). Thank you very much for your support over the last (TIME FRAME) years. It was a pleasure to work with you on (PROJECT) while I was part of the (COMPANY) team. (PERSON) will be handling your account beginning (DATE). You may contact her at (email@COMPANY.com) or by phone at (NUMBER). If I can be of any assistance during this transition, please don’t hesitate to let me know.
If you have any additional questions or wish to keep in touch, my personal e-mail address is (email@example.com) and my phone number is (NUMBER). I wish you all the best in your future.