I have read that Timur Lang was a conqueror who attempted to restore the Mongol Empire. But why did he attempted to do that? Was he a decendant of Genghis Khan as he used to say?
Timur was a warlord of (partial) Mongolian descent from Iran, one of the four major divisions of the Mongolian Empire. (The other three were Russia, Kazakhstan, and Mongolia-China.) As such, he was "more royalist than the king," regarding the Mongol empire, in his attempt to restore it to its former glory, under his own rule.
His first step was to unite the former Mongolian Iranian territory under his rule by the 1390s (although he also advanced further into India than Genghis Khan). Then he conquered much of "Mongolian" central Asia (Kazakhstan, etc.) Finally, he planned to make war on the former Eastern Mongolian empire (Ming China and the Mongolian homeland), but died in 1405 before he could do so.
It's possible that he was descended from Genghis Khan (many people in that part of the world were), but that is unclear, or even disputed. Only a DNA test (if it were possible to run one) could tell for sure, because there are no records of most of Genghis Khan's affairs, and the resulting offspring.
At the time conquering yourself an empire was seen as a legitimate ambition if you could do it. Invoking Genghis Khan was a way to gather supporters and frighten opponents: getting people to surrender is easier than fighting them, and means you can tax them.
He was not a member of the Borjigin clan, but may have been descended from Genghis Khan anyway. There's no way to know now.