The newly-promoted Championship side, who beat Sheffield United in the play-off final on penalties, secured the player on Friday who has experience working under the Town coach
The 23-year-old played under Grayson when the coach was in charge at Elland Road and made the switch between Yorkshire clubs for an undisclosed fee on Friday.
The central midfielder, who made 43 Championship appearances for Leeds last season, is well-known to the Terriers manager and he believes the transfer shows the intent of the newly-promoted club.
“Adam is a good footballer,” Grayson told Huddersfield’s official website. “He handles the ball and has a good energy about him.
“He had a great season last year and scored some important goals. We're delighted to have got him.
“He's a player I admired when he was at Carlisle and Manchester City and we're delighted to secure his services to this football club. I think it shows our intent in the level of players we want to sign.”
Clayton netted six times for Leeds last season in the league and Grayson wants his side's goals to come from other areas aside from up front.
“I think we need to score goals from all areas of the pitch,” the coach admitted. “We can't just rely on the strikers to do it; we need the wide players, the centre halves and the midfield players to chip in.
“Now we have players who are capable of getting into the box and that can contribute in set-piece situations.
“We can't simply rely on Jordan [Rhodes] to get us another 40 goals this season, just as we can't rely on the other strikers.
“We need goals from all areas of the pitch and Adam has that capability, but he also has other sides to his game that I've seen when I've worked previously with him.”
Sean Scannell, Paul Dixon and Oliver Norwood are Grayson’s previous three signings before Clayton, and the Huddersfield boss hopes the new faces can help propel Huddersfield up the league from the very first whistle.
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“I think things are moving along nicely in all departments,” Grayson continued. “You have to strike a blend and get the partnerships right all over the pitch and we are still working hard behind the scenes to get one or two other things done.
“We're looking to keep improving all the time, but we're getting the right ingredients for a really decent squad.
“With losing Damien Johnson and Tommy Miller, we did feel we needed players in the midfield area and we've brought two young, hungry players in with Adam and Norwood.
“It's also great that they both have experience of the Championship; Adam played all last season in the Championship and Oliver played half a season for Coventry, so we will only benefit from that experience.”