You want DC, not AC. Typically anywhere from about 400V to 1500V DC, depending on your GM tube. 900V is probably the most common defacto standard for US-made Cold War era Geiger counters and GM tubes.
400VAC driving a single Cockroft-Walton style voltage doubler stage might make sense, though.
Look at these schematics for some ideas:
http://mightyohm.com/files/geiger/geiger_sch_fixedR5R6.pnghttp://dlnmh9ip6v2uc.cloudfront.net/datasheets/Sensors/Radiation/Geiger_Counter-v30_2.pdfYou can either go for a boost converter sort of approach using an inductor, or a transformer-based approach. You can use common, cheap transformers that you can find cheaply or recycle, such as 8-ohm speaker drive transformers, or 600-ohm telephone line isolation transformers from modems for example.
Another approach that is sometimes used is to use a small CFL inverter board, which is a cheap surplus thing such as you'd find in a scanner or laser printer or monitor or perspex-window case-modded PC to drive the CFL lamp.
This might typically output, say, 400-500V AC. You'd need to remove the series output capacitor after the transformer, add a single-stage voltage doubler to the output, and regulate the input voltage to get the output voltage that you need.