There are basically two obvious ways to go about this, do it all by yourself or under some professional guidance.
If you are a seasoned boulderer, you may prefer to do it without any special guidance. I mean even if you are new to Hampi. A little bit of homework and betas collected from your friends who have been to Hampi is adequate you to start with.You will start bouldering in Hampi without any hiccups.
In general you neither have to take any special permission nor pay any fees for bouldering in Hampi.
The second option is to seek professional guidance. This is recommended especially if you are a novice.
There are many agencies and bouldering professionals offer their services at Hampi. Some are mentioned below
A third option is of course is to join a group of other boulderers.
Thimma Climb is a local climbing setup run by local climbing guide. Also rents/sells crashpad,chalk bag, magnesie/chalk, Tape etc. They are on the Virupapur side.
Tom & Jerry Climbing Shop Rental of climbing equipment, climbing trips, sessions at Hampi. They are located in Virupapur Gadde area.
Connect with 'Reachingout' (climbinginindia.com) for what options are available for the season.
Check out the website of Manav , voluntary organization known for their activities in adventure sports ( www.manavngo.org ) .
Nature Admire is a private agency offers training in adventure sports. Check with them if they have any Hampi specific program coinciding with your Hampi plan ( www.natureadmire.com )
There are a few guide books and other resources available for self help. Checkout the 2014 published Bouldering Guide: Golden Boulders available both in English and German.
Virupapur Gaddi Guesthouses
You need to get on a coracle/local boat to reach Virupapur Gaddi. That's a good introduction to this backpacker hideout in Hampi.
Hampi on a shoestring!
The very virtue of Hampi supports your idea of a shoestring budget tour. Certain words complement so well, it seems Hampi and shoestring are one such pair!