Clarke was speaking to Sky Sports News following a broadcast which uncovered several incidents of racist chanting, with Millwall fans persistently subjecting Leeds' El-Hadji Diouf to racial abuse at the Den in November.
Footage appeared to show the Senegal international alerting the referee, Mark Halsey, and making stewards aware of his allegations but Millwall insisted no complaint of racist abuse was made by the player.
The report was inspired by Marvin Sordell, the Bolton striker who said he was called a "slave" by Millwall fans last year and is one of the few footballers to speak out about his experiences.
Clarke, who was at the Millwall v Leeds match, told Sky Sports News: "I was grateful to Sky for shedding some light on a very dark area in football. Anybody who has been associated with football for years knows that things are getting better but they are nowhere near where they need to be.
"To have our players subject to vile and filthy abuse deserves nothing better than the culprits being arrested and charged with the full weight of the law.
"We obviously have to stamp out racism. No right-thinking person could in any way condone the sort of terrible behaviour we saw on your programme.
"But what chilled me even more was not the racism but the reticence of black players to feel comfortable in reporting it.
"If we created an environment which doesn't support black players in confronting this issue then we've done something terribly wrong.
"So we've got to wrap ourselves around our black players, give them our support to be able to report such episodes and then deal with such episodes extremely severely.
"I've been working with Herman Ouseley (chairman of the Kick It Out campaign) and (PFA chief) Gordon Taylor behind the scenes, meeting one-on-one with black players, senior pros, managers, who, to be frank, don't want to go on camera and don't want to put their head above the parapet but believe things need to be done.
"So we need to create a culture within football which rather than make the person pointing out the problem the villain, that they get the support they deserve to be able to point it out to us so we can do something about it. We need to support the victims here.
And when asked what steps the Football League would be taking to stamp out racism, Clarke responded: "By making sure that we use every tool in our armour.
"We will use CCTV, steward statements, players statements, to hunt down every racist at every football ground and make sure they never get into a football ground again, plus be charged by the police."
Millwall released a statement on Monday night in which they said they were "appalled" by the footage they had witnessed.
Allegations of abuse against Diouf initially surfaced online directly after the match, with Millwall claiming an investigation carried out at the time resulted in there being no case to answer to.
Millwall chief executive Andy Ambler said: "That day against Leeds obviously there was an allegation of abuse made online initially. We investigated it straight away.
"We interviewed stewards, police, players, we looked at the TV footage too as it was live on Sky.
"After speaking to the player who had the alleged abuse at him during the day, he said that he heard no abuse on the day so that was basically at that point the end of the investigation - although we're going to open it again now."
Ambler added: "Clearly that behaviour is unacceptable in any football ground in the country and at Millwall we will ban the individuals for life and we hope that Sky will pass the footage to the police so that further action can be taken."
Meanwhile, the chief superintendent of the Metropolitan Police's Public Order branch, Mick Johnson, says his force is working closely with football clubs at tackling racism.
Sordell, who represented Great Britain at the London 2012 Olympics, has faced racial abuse on social media websites and claims he is not shocked that such things still occur.
He said: "We're a developed, multi-cultural society. It's surprising it can still go on - but it doesn't exactly shock me.
"We can't be silent about it. We need to make people aware that there is a problem going on and only when people are aware of the problem can it be sorted out."
Football Association chairman David Bernstein said he was treating allegations with the utmost importance, telling Sky Sports News: "These issues bother me intensely. They have become top of my agenda in terms of moving these things on and combating these things.
"The awareness of it could not be higher."