Generally speaking, you will not see much improvement from having multiple tempdb files, except under specific conditions that are fairly rare.
When you are allocating an extremely high number of temp tables, there can sometimes be contention on certain internal structures that have to be updated for each temp table. Since these tables are on a per file basis, you can reduce contention by having multiple files. For a more complete explanation of this, you should read Paul S. Randal’s explanation of this:
You might want to have multiple tempdb data files if you are going to put them on multiple volumes, but that does not sound like what you are planning.
The performance improvement that you get from having tempdb on SSDs will depend a lot on your application. If the application currently exists on another server, you might get some idea of the expected improvement by looking at the amount of IO and PAGELATCH_XX waits on tempdb that you have currently.
As others have suggested, you might get more performance improvement from using SSDs for application database data and log files.