You can copy your client settings through "manage profiles" in the launcher, if you want to do so. Everything else (creating and patching of the client) is done by the launcher. Just select "Singularity" in the server drop-down menu and then login with your account.
With the EVE launcher (for both Windows and Mac) it is very easy to connect to Singularity:
The account state will be the same as it was when the mirror was taken. The account needs to have been created before the time when the current mirror was taken. In order to access the Singularity server you need to have an account on Tranquility. If you have more feedback please post it in the Test Server Feedback Forum. They will most likely be changed and tweaked further. Hint: Please keep in mind that most new features on the server are still in development and not set in stone. For testing the performance of the server and client with many players at the same place (especially fleet-fights) there are periodical mass testing events. For this purpose, the Tranquility database is "mirrored" (copied) over to the Singularity server once every few months (or more often, if needed by the CCP Quality Assurance department). Its primary function is to provide a test environment as close as possible to the live environment of Tranquility. Test servers have a very strict set of rules. Singularity (Sisi) is a public test server for EVE Online that is used by both CCP developers and EVE players to test changes and new features before they are released to the Tranquility server.