I have a Vox AC4TV amp, and I am happy with it, but I want it to be a tad louder.
If I hook it up to a bigger Cabinet will I get more sound out of it?
And what cabinet would you recommend if my budget can stretch up to $300?
I don't mind used stuff.
More sound yes... but more sound doesn't necessarily equal louder...

Musician's friend is a good resource... look up reviews on cabinet that matches your output impedance..
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Think the Celestion VX10 is around 95/db sensitivity, so using a cab with a speaker that is more efficient, say a V12 at 100/db, will cause a fairly significant volume increase.

Not sure the V12 is suited to the AC4, maybe something with a G12H30 in.
Check out the Dime cabs, good reviews and great price. 412 is in your price range.
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Quote by Strats&Cats
Ever thought of selling the AC4TV and picking up a used AC15 or something?

Not really. . .
I rather buy a cab, and maybe mod my AC4TV.