Step 1: Get ready with NEO-GUI

Download NEO-GUI (Desktop) and then, open neo-gui.exe and wait for it to sync to the network.

Your sync will be completed when the left number of blocks is equal to the right (red rectangle from top figure). While waiting for it to sync, you can create a new wallet for your NEO. Go to Wallet, click on New Wallet Database.

Click on Browse, and save your wallet with the format .db3 by default. Then key in your password. This password is used to open your wallet database. Note: Backed up your .db3 file. After that open your wallet database.

Now you can create new address for your NEO. Your address will showed up under Standard Account (Figure below).

Right click your address and click 'View Private Key', write it down and keep your private key safe. Now you can deposit NEO to this address.

Step 2: Get NEO

You can get NEO from exchanges such as Binance, Bitfinex, Bittrex, HitBTC and many more. Myself using Bitfinex, HitBTC and Binance as the withdrawal fees is zero(Fees may varies from time to time).

Now that you have NEO, you can transfer them to your NEO-GUI address. Remember that only transfer whole number of NEO to your address.

Step 3: Claim GAS

You can view your GAS by clicking on NeoGas Claim from Advanced (red rectangle in figure below).

By default, you cannot claim your GAS that showed in Unavailable. To claim your GAS, you need to transfer your NEO to the same address.

Click on Transaction, then Transfer. From the Transfer layout, click the '+' (add) button. It will bring you to Payment layout. Asset: select NEO, Pay to: paste the same address that show up on Standard Account. Amount: total NEO. Then click OK all the way.

After that, go back to NeoGas Claim, now you can click Claim All to claim all your gas. Your GAS will show up under NeoGas.

That's how you claim your GAS.

Extra:

Claim GAS from NEO using NEON wallet
Check your NEO address here
NEO to GAS calculator