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I'm looking for the number of soldiers the Mongol Empire had at the time of Genghis Khan. What percentage of them were ethnically Mongolians?

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In this period, the Mongols had a nominal strength of around 100,000 to 130,000.

According to the Secret History of the Mongols, Ghengis Khan had an army of 105,000 strong by A.D. 1206. This number grew to 129,000 by A.D. 1227, according to Rashid-al-Din_Hamadani in his Jāmiʿ al-tawārīkh. Of the latter figure, 27,000 were new units raised from Manchuria. If we assume the rest were "Mongolian", then:

Without the forces added later, Rashi al-Din has 102,000, or 3,000 less than the Secret History's total. This difference could be explained by units left to guard the western regions. It therefore seems that the total number of "Mongol" soldiers was quite constant from 1206 to 1227.

- Sverdrup, Carl. "Numbers in Mongol Warfare." Journal of Medieval Military History 2010 (8).

However, these figures are nominal estimates based on the tümen unit.

[C]ontemporary soruces often quote the number of army units or the number of commanders of tumens, for example. However, though theoretically a tumen consisted of 10,000 men and a commander of a tumen had 10,000 men under his command, this was rarely the case, and in reality numbers could be very considerably less.

- Lane, George. Daily Life in the Mongol Empire. Greenwood Publishing Group, 2006.

As is usually the case everywhere else, not every individual units would have been at full strength. They certainly would be diminished on campaign, by both natural attrition and casualties in fighting. Moreover, some would have had to stayed behind at home. Thus,

The effective total of the Mongol army was likely 70-80% of the nominal total. The percentage of effectives should have been high when the army set out from home territory with a gradual decrease as time passed.

- Sverdrup, Carl. "Numbers in Mongol Warfare." Journal of Medieval Military History 2010 (8).

Therefore, Genghis Khan probably fielded no more than 100,000 soldiers in total at any one time. This excludes conquered subjects pressed into service as infantry meat shields, though.

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Just before Ghenkis Khan died, he invaded Western Xia with 180,000 soldiers, according to Wikipedia.

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