You definitely need some for of specialised instrument to do this whether it is home built or commercial. You can make basic measurement (down to about -80dBV) with something like the Ferrograph RTS2 test set (see link below) which can still be obtained for a reasonable sum on eBay. (the Ferrograph will also measure distortion at 1KHz).
http://www.morphet.org.uk/images/ferro/ferro-rts2man/cover.gifTo make accurate, repeatable noise measurements you need 3 things:
1. Low noise gain
2. An appropriate filter
3. Appropriate meter ballistics
Opinions differ a lot on 2 and 3 with the marketing guys preferring 'A' weighting filters and rms ballistics as these give the most favourable readings. Manufacturers of professional audio measuring equipment use different filters and quasi peak ballistics because they are of the view it gives readings which more closely represent what we actually hear. I use a Lindos Minisonic MS10 which can accurately measure noise down to -90dBu and conforms to IEC268 (CCIR 468) weighting and Quasi-Peak dynamics and has balanced inputs. The basic unit cost 400GBP but it also does frequency response and distortion measurements too.
http://www.lindos.co.uk/ms10Whatever route you take, making noise measurements is not easy (especially of prototype equipment). It is amazing where noise gets in (from soldering irons, flourescent tubes, laptop PSUs etc) so you need to take special care to ensure what you are reading really is noise.
Cheers
Ian
Edit: Here is a picture of my tube noise test jig. It fully encloses the tube under test in a screened box. Heaters wires go straight out the back away from HT and signal wires. The jack plug goes straight into my Lindos.