Looking to build a system for under $5k

Good advice to audition some Vandersteens or other speakers first. Then choose an amp that is a good match for your speakers; proper power rating, features, and tonal characteristics that are right for your taste.
An integrated amp would be the way to go to fit your budget. Some integrateds have a phonostage built-in for a TT/cartridge, or you can use a dedicated phono preamp.

Thanks so much! Audio Connection is already on my radar as a source to check out!

You may want to prepare for the future, (which is music streaming), by looking at an Oppo spinner which plays CD’s and can accept streaming services.

In general, I'd advise thinking about the long term. Ask: will I always be in the upstairs room with smallish dimensions? This might change your approach somewhat.

For what you describe, I'd go this way. Prices are approximate, presuming you'd shop around some for slightly used/demo stuff:

-Rogue Sphinx integrated w. phono stage: $1200.00
-Sony HAP-Z1ES digital server (you'll stop using CDs forever, and you'll be able to stream Spotify too): $1800.00 (or less)
-Vandersteen speakers that you mentioned: $750.00 (?), because you like them. Or get smaller monitors (ELAC?), many possibilities for <$1000.00
-Hsu Research Subwoofer (various models available): $500.00. Subwoofer is the best way to get real bass in a small room.
-Use the stock power cords for now.  Upgrade them later or not.  Use the rest of your budget for inexpensive interconnects and some speaker cables.

good luck!

I suggest a used solid state amp to drive your Vandersteen speakers.  They are fine speakers and you can buy now if in a hurry.

for the amp, maybe a Hafler?

I also like the Oppo player for this - lots of used ones as people upgrade.  You may be able to get by with no pre-amp.  

Do NOT waste your money on any sort of expensive cables.

Also take a look at Cambridge Audio equipment; Rotel; and similar.

Marantz and Yamaha make good receivers if you decide to go that direction.

Try to save some money out for room treatments - much of which can be DIY - I've posted some links before on other threads.

One advantage is your room is non-square...