Any time excluding rainy season, that is a bit risky. Feb March April and october november may be considered as the best as these months are safe from extreme cold and extreme rain.
Gangotri Gaumukh Tapovan Trek Area: Garhwal Himalayas
Duration: 10 Days approx.
Altitude: 4463 mts /14638 ft
Grade: Moderate to Challenging Season: May - June & Aug - Oct
Go in June. It is a pleasant time to trek Gaumukh Tapovan. you will find crowd during this time of year. and also sky stays clear so you can enjoy snow capped himalayan peaks clearly in front of you.
Most will recommend you May to June. But in my opinion October is best. Why? During May June, it's yatra season in Uttarakhand and on your way, you will stuck in traffic. And due to high influx of yatris, getting taxi, or hotel is little bit pricy. That's why I prefer October. Less Yatri during that time, open sky, and peaks are properly visible. Check with Himalayan Dream Treks if you are looking for Gaumukh Tapovan Package.