I'm saving this post and the next one for reporting on the tests and methods I use with each of the champions.
Test 1: I started the test with 21/9/0 masteries, Mag Pen Reds/Armor Yellows/Flat Ap Blues + Quints, Exhaust/Flash
This guy's clear time is really, really slow. At 10 minutes I only had about 50 cs, was level 7 (almost 8). Something to keep in mind is to be standing right next to Vilemaw when you start to drain him or he'll pull you and just crush you while you wait for your cooldown to reset. I had Sorc Shoes/Haunting Guise/Amplifying Tome and it was close but I only got him to 30% health before he killed me. Even when I ulted into him with the same items I only got him to 20% bbefore dieing. I went back to farming, got level 9 and made my Tome into a Hextech Revolver. I went back and killed Vile easily, never dropping below half health.
Test 2: Using the same setup as before I did my best to get as high a level as I could by 10 minutes. At 10 I was level 8 with about 57 cs and I had Sorc Shoes/Hextech Revolver/Amplifying Tome/Ruby Crystal. It seems the Revolver was the trick that did it and I barely ended up beating Vilemaw. If you have an ally helping you just a little bit, or who tanks one or two hits you should be able to beat him easily. Alone thought Fid has one of the slowest clear times; it took me about 35-40 seconds to solo Vile. Also I ran the test a few times, your success depends on if Vile straight attacks you a lot or tries to hit you with his AoE slam. If he is just aa'ing you then you're screwed but if he does his slam it gives you time to heal between attacks. If you can get to level 9 though you can solo Vile easily, though it is still a slow clear time.
Test 1: On initial test with Jax, 21/9/0 masteries, AD Reds/Armor Yellows+Quints/Mag Resist Blues, Smite/Exhaust
Jax starting clear speed is a bit on the slow side but quickly picks up by the time he is level 4. At 10 minutes I had 70 cs, was level 9 and had Boots/Spec Lantern/Sheen/Long Sword/1 Health Pot. During the fight with Vilemaw I used the pot and my ult's active but I killed Vilemaw in about 20 seconds and still had a quarter health left.
Test 1: On initial test with Shyvana, 9/21/0 masteries, AS Reds/Armor Yellows/Mag Resist Blues/AD Quints, Smite/Exhaust
I took farm only from my jungle and after level 4 the enemies' wolves when smite was up. At 10 minutes I had 70 farm and was level 9. I had Boots/Spec Lantern/LifeSteal Scepter/Ruby Crystal/Emblem of Valor/1 Health Pot. By the time I got it to half life (smite included) I was almost dead (health pot already used) and I backed off and continued farming.
At 13 minutes I was level 11 with 100 farm and Boots/Spec Lantrn/LifeSteal Scepter/Aegis of the Legion. At this point I could solo it (with smite) but it was still a narrow victory and a bit luck based on Spec Lanturn's passive.
Test 2:Second test with Shyvana, same Runes/Masteries/Spells
At 10 minutes I had 73 cs and was level 9. I had Bers Grieves, Spec Lantern, Recurve Bow, Long Sword, 1 Health Pot. I still couldn't solo it at 10 minutes with this build by my dps was much higher and I took it down to about 40% health before I had to back off. With help from your top lane ally and this build you should be able to get Vilemaw around 10:00-10:30.
Test 3:Same setup as tests 1 and 2. This time I waited until 11:30 to take on Vilemaw. I was level 11 with 93 cs and had Boots/Spec Lantern/Wit's End/Long Sword/1 Health Pot. It was a narrow victory that relied on the Lantern passive but I did solo Vilemaw. If your top lane ally helped you at all it should be a pretty easy victory.