I have already been there an year ago.Budget will not exceed 5k. The trek is bit long and you cannot reach the top in one go. The second half is bit dangerous.In all it took us around 14-15 hr walk to reach the top.
You can start from gulbarga... it's not a one go trek if you are not a regular. I went there last may... booked it thru TREK THE HIMALAYAS... it was a 3day trek... budget was around 7-9k
The trek trail being of easy to moderate level, it requires less fitness levels and is suitable to every age group and people as compared to other Himalayan treks.
https://www.thesearchingsouls.com/bhrigu-lake-trek/