Are there any personality changes when cancer starts to metasticize?

Can be so. It can be a complex question, since chemotherapy and radiation can have side effects, and mood disorders such as anxiety or depression can produce some treatable personality changes. If there are mets to the brain, certainly personality changes can develop. Experienced medical or health clinical psychologists can sometimes add perspective to medical evaluation that would be done by other docs.
"cart before horse" I think it is fair to say that the fear of cancer metastasizing is likely to affect one's personality (although, theoretically, mets to the brain may cause personality changes). I think that the greatest challenge for any cancer survivor is how to live with that fear hanging over one's head. Many of my patients have benefited from counseling to learn the best coping skills.
Can be. Depression is common. If there is pain, it certainly can affect mood. Metastasis to brain can manifest with altered mental status and/or mood changes. Liver metastasis can also do the same.
Personality changes. This depends on the areas of the brain where tumors metastasize.