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The Kauravas are biological descendants of Vyasa, but the Pandavas are not, as I'll explain below. But first, let me explain above tree.

Shantanu, king of Hastinapura many generations before the Pandavas and Kauravas, married the goddess Ganga and had a son named Bhishma. But then Ganga abandoned Shantanu, and Shantanu later fell in love with a woman Satyavati. But Satyavati's (adopted) father Matsyagandha, the chief of a fisherman tribe, wouldn't consent to the marriage, because he was afraid that Bhishma and his descendants would rule Hastinapura rather than Satyavati's children. So Bhishma made a vow that he would never become king, and that he would remain celibate so that his descendants could never claim the throne. (It's because of that terrible vow that he got the name "Bhishma" or terrible.) 

Then Matsyagandha agreed to the marriage, and in due course of time, Shantanu and Satyavati had two sons, Vichitravirya and Chitrangadaa. When Shantanu died Chitrangada became king, but he soon died after a confrontation with a Gandharva because the two of them had the same name (yes, really!). (YOU CAN ASK A DIFFERENT QUESTION FOR THIS STORY)

Now coming to the next generation kids 

When Vichitravirya became of marriageable age, Bhishma heard that the king of Kashi was holding a Swayamvara for his daughters Amba, Ambika, and Ambalika. So Bhishma went to the Swayamvara kidnapped the three princesses. 

But, he released the eldest princess Amba, a decision that would later have devastating consequences. (Again an interesting story, another question in needed for the same)  

In any case, Vichitravirya married Ambika and Ambalika, but almost immediately thereafter he died of a disease. So now the kingdom faced an uncertain future because Vichitravirya and Chitrangadu were dead, and Bhishma could not have children, which meant there was no prospect of heirs to the throne. 

So Shantanu's widow Satyavati sent for the sage Vyasa, a son she had with the sage Parashara before she met Shantanu, and told him to father sons with the two queens.

Now Vyasa, being an ascetic, didn't pay much attention to his appearance, so the queens were quite hesitant to have children with him. As described in the Adi Parva of the Mahabharata, Ambika closed her eyes the whole time, so Vyasa got angry and put a curse on her that she would have a blind son, Dhritharashtra. And Ambalika was pale the whole time, so Vyasa put a curse on her that her son Pandu would have leprosy. 

(On the other hand, Ambika's maidservant was eager to have a child with such an illustrious sage, so Vyasa fathered a child with her as well, Vidura.)

So yes, this does mean that biologically the Kauravas were the grandchildren of Vyasa, since they were the sons of Dhritarashtra and Gandhari. 

But that doesn't mean that the Pandavas are the grandsons of Vyasa as well. 

You see, although the Pandavas were the sons of Kunti and Madri, the two wives of Pandu, they are not in fact Pandu's biological children. Pandu had incurred a curse that he could not touch any woman with desire. 

So Kunti used a special mantra she had acquired from the sage Durvasa which allowed you to have children with any god. So Kunti used the mantra to have three sons - Yudhishtra from Yama the god of death, Arjuna from Indra, and Bhima from Vayu the wind god. And she taught the mantra to Madri who used it to have two sons, Nakula and Sahadeva, with the two Ashwini Kumaras.

So to sum up, the Kauravas had the blood of Vyasa, and thus some small amount of blood from Shantanu's wife Satyavati, albeit not the blood of Shantanu himself. The Pandavas, on the other hand, didn't even have the blood of Vyasa, being the sons of gods. 

Ironically, the only living person at the time of the Mahabharata war who had blood from the actual royal family of Hastinapura was Bhishma, the one person who couldn't be king or have children. So in the strictest sense the bloodline of Shantanu ended with the death of Bhishma at the end of the Mahabharata war.