Hi Hattond, I shall bet you have a desktop computer or a laptop you are running with a charger attached? This has resulted in an "earth loop" (aka ground, hum loop).
You were in the right area when you suggested a "DI* box" but, as told, not what you want. Try one of these:
AV Link Ground Loop Isolator: Amazon.co.uk: Hi-Fi & Speakers
Ok, not Soooper-fi! But actually not that bad and they do the basic job of breaking the shield circuit perfectly well and that is not a lot of money for a "diagnostic" tool?
Note also above right? Two RCA to 1/4" adaptors for £2.19? These can be used with your existing RCA-RCA cable to output the F'rite. That Hosa cable is very fine I am sure but even here in Blighty we can get one for 1/3rd the price!
If that isolator does the job there are much better ones at a lot more money of course. Look up Art and if in EU Orchid Electronics. Or, there is a trick you can do which involves modifying the audio cables which carries no quality hit.
*A "DI Box" is generally a device that accepts the input from a passive electric guitar and presents said guitar with a fairly high impedance. It outputs at low, generally less than 600 Ohms impedance, balanced, earth free on an XLR at microphone level. Almost all DIs have a 'slave' out jack to feed an amp.
This terminology is for ever confused with the device YOU need which is a 1:1 transformer isolator, absolutely NOT suited to electric guitar.
Dave.