In general, women who are at increased risk are advised to check their own breasts regularly in the week after each period, and report any changes. From the age of 50, they will be offered NHS mammography. There is controversy about whether younger women in a higher risk group should also have mammography.
The breast tissue is dense so the results may be technically inconclusive, and repeated X-rays themselves may be a risk factor. Women with a seriously increased risk may be offered regular mammography from the age of 40.
Even with a positive family history, the odds are still against you developing breast cancer.