The hardest thing about cleaning a silver bracelet chain is getting in between where the links join and into the centre of each individual link.
I have found that soaking the chains first in a bowl of hot (not boiling) water with a couple of drops of a mild dishwashing detergent helps to loosen most dirt and grime. It is a good idea to leave them in for 10 or 15 minutes and swish them around every couple of minutes while they are soaking.
If you have stones on your bracelet check that water won’t harm them before doing this. Jewellery with opals for example should never be allowed to get wet.
When the soaking is finished brush gently between the links of your silver bracelet with a soft bristled toothbrush, starting at one end and working down towards the other. When you are done rinse the silver bracelet or necklace well under very warm running water and dry thoroughly.