Can Professor X read the mind of Wolverine even if he has an adamantium skull?
Yes, Wolverine's adamantium laced skull did not prevent his mind from being read by any competent telepath. Charles Xavier, Jean Grey and Emma Frost were all able to read Wolverine's calm mind despite his adamantium advantage.
However, when Wolverine was enraged or in a berserk frenzy his mind became much harder to read and control which made it possible for Wolverine to block access to his mind during that time.
Early in their relationship, Xavier found it quite difficult to access Logan's memories which appeared to be layered with psychic limitations imposed by an unknown source and periods of memory loss. It took Xavier many years to help Logan reconstruct his memory. This reconstruction helped Xavier to be able to read Logan's mind later in their interactions.
After his bruising encounter with Cyclops and Emma Frost, Professor X is forced to revisit the biggest challenge and the biggest failure of his career, Wolverine, when the feral mutant asks for Charles' help in freeing his son from the clutches of the Hellfire Club. As the two search for Daken, Wolverine reveals that when he first joined the X-Men he attempted to assassinate Xavier due to some unknown programming. In response, the Professor broke Logan's mind and rebuilt it so that any and all programming he received was forgotten. Revealed in "X-Men Legacy" #217.
Wolverine or any other mutant/meta-human who posses the ability to regenerate has a small natural resistance to telepathy due to there brain cells healing to the way they were before they were altered. This effect is not potent enough to fully stop any good psychic from accessing and changing anything.