Williams is currently preparing to lead his side in their Euro 2016 semi-final fixture against Portugal on Wednesday with the hope of securing a spot in the final of the European showpiece.
Speed, like Williams, captained the Welsh national team and went on to win 87 caps over the course of his international career before transitioning into managing Wales.
His tenure did not last long however, as he tragically took his own life in November 2011 having overseen only a handful of games.
"I was thinking about him on the plane coming over," Williams told reporters. "I hope he'd be proud of us. He saw us grow up, and you see what we've achieved today.
"He's always in the back of our minds. When we achieve something, we think about him."
Wales manager Chris Coleman, who succeeded in the national team job, acknowledged the foundation that the former Newcastle United man laid for him.
"People talk about his legacy as a manager, but it's hard to have a legacy in 10 games," he said.
"Speedo's legacy is 85 caps for Wales, and maybe over 50 as a captain, plus his 10 or 11 games as a manager, in which he showed great promise as a young manager.
"And then what happened, happened, unfortunately. He's not just a Welsh but a sporting icon. But I don't need a game of football for me to remember Speed.
"I think about him every day, not just because I'm at a tournament. I don't need Wales football to remind me of my friend. He could be sitting here where I am, enjoying what we're enjoying. Unfortunately that was taken away."
