Celtic are closing in on a fifth-straight title, but the Gers are keen to put a stop to that next season.
Rangers secured their promotion to the Premiership on Tuesday and have aspirations to compete at the top of the division, according to Murray.
But the businessman is wary of the misery suffered by the Light Blues' over the past four years and is keen to avoid a repeat situation.
Following the culmination of what Murray refers to a 'phase one' - getting out of the lower leagues - the 51-year-old told the Record: "Phase two is to get the club back to competitive level with a view to winning the Premiership title and we want to do that as soon as we can.
"I am not going to make outlandish promises and say we are going to win the league in the first season but we will obviously try and make the team as competitive as we can as quickly as possible.
"I don't think you can put figures on that. It is not always about throwing money at the situation. First and foremost we don't want to put the club into financial risk.
"We have just been through four years of misery so we are not going to do it again.
"Obviously the benchmark in Scotland is Celtic.
"With a combination of investment, a good management team which we believe we have, a good recruitment structure which we have put in place behind the scenes and also a progressive youth structure, putting that together, there is no reason why we can't compete pretty quickly."
Rangers only need to look at Hearts' example to see how spending frugally can still get you to where you want to be in Scotland's top flight.
The Edinburgh side were promoted as champions last season, but only two of their 12 signings this season - Gavin Reilly and John Souttar - were fee-paying ones.
Robbie Neilson's men are currently third in the Premiership, though Rangers will be hoping to do better than that themselves.