Councillors are elected for a term of four years; the next local elections will be in May 2018, so you have plenty of time to prepare!
There is a lot of information about the Council and Cllrs on the lewisham.gov website that you should look at carefully.
Do be aware of the political make up of the current Council which is essentially Labour with a single Green Party Cllr.
I cannot recall any independent councillors getting elected over the last 40 years in Lewisham, although many have stood as candidates. In some, more rural parts of the UK, Independents do get elected, but they often have very right wing views. They often cannot take control of a Council because they have such differing views!
You may find it very hard work to stand as an Independent, because you will NOT have an established team of party workers to support you, and you will have to find ways of contacting your electorate (many thousands of households) with leaflets and personal calls.
Worth a try?