I am supposed to analyse the Land use/Land cover of some particular area, taking into consideration the various atmospheric and land surface conditions- satellite image of which month is best suitable?
Generally the driest month is the best for differentiating between land uses, end of the dry season if there are a number of dry months. Make sure if you are comparing between years that you check rainfall data, you don't want to be comparing dry and wet season.
The original post has already acknowledged accounting for atmospheric effects (clouds). That said, I think it's necessary to be more specific about the study area. India is a diverse country, covering many latitudes, ranging from tropical conditions to the Himalayas. In an area with diverse vegetation, you might need multiple months (i.e. leaf-on and leaf-off) to do an adequate classification. In areas without deciduous vegetation or major seasonal crop variations, one month might suffice. In other words, it's really impossible to generalize the answer for the entire country of India.
as you are analysing land use, it may be useful to first determin the main abiotic limiting factor/s to the success of the landuse intent. Then correspond the seasonal time period in which the abiotic stressor occurs; this will help target a period for investigation.