do tumour cells begin with abnormal characteristics?
Cancer cells don't start to become cancerous at a specific stage of the cell cycle; you will find that while uncontrolled proliferation is a hallmark of cancer, different cancers acquire alterations in different phases of the cell cycle. BRCA-deficient cancers for example have a compromised G2-M checkpoint [1], while Rb deficient cancers have a compromised G1-S [2] checkpoint in the cell cycle. 

The cell cycle is simply a property of proliferating cells and the same broad phases of the cell cycle are universal to both normal and malignant cells. 

As for when abnormally growing cells actually become a cancer - this has nothing to do with phases of the cell cycle, and everything to do with the ability to break through the basement membrane of the original site (indicating the potential to be invasive/spread), because the ability to invade is a hallmark of cancer [3]

References 

[1] http://cancerres.aacrjournals.org/content/67/13/6286

[2]  https://www.ncbi.nlm.nih.gov/pubmed/16936740

[3]  https://www.ncbi.nlm.nih.gov/pubmed/21376230