I have used SQL Clr procs and triggers with very good effect to help database performance in some cases. There are situations where your TSQL logic is just going to run slow (full text search without FTS indexes), and there are some things you just can't do in TSQL (lookup a DNS address, map reduce, image processing, OCR, unzip a file, complex algorithms, etc).
I'm considering it because I have to look at some data, run it through
a bunch of procedural logic, update, and then get it back to the
client. My thinking here is to get as close to the DB metal as
possible, and use that hardware to make it happen quickly
But that is not a scenario where I would recommend using them. The other poster is correct that set theory almost always has a way to do what you want.
SQL CLR is not closer to the metal, you are running a full instance of .net clr inside your database. If you have none right now, adding one will add a lot of RAM usage for that one proc. If you have 20 then the cost is not so bad "per proc".
If you are trying to do all these calcs to get the data back to the client, then just do it on the client. The client usually has close to 100% of their CPU available to them. The server will never have anywhere near that for one user.
If you have truly central logic or complex business requirements that you don't want to put on every client, put that on the server. Don't do simple CRUD operations in SQL Clr, it doesn't buy you anything usually.