What is the speed of the fastest moving body in our solar system?
The maximum speed of an object that orbits the Sun at a certain distance $r$ is known as the escape velocity: $$ v_\text{esc} = \sqrt{\frac{2GM_\odot}{r}}, $$ where $M_\odot$ is the mass of the Sun. If the object would have a greater speed, it would eventually leave the solar system. So I'd say that the absolute maximum possible speed of any object in the solar system would be the escape velocity at the radius of the Sun $R_\odot$: $$ v_\max = \sqrt{\frac{2GM_\odot}{R_\odot}}, $$ which, as you can find in the wiki article, is $617.5\;\text{km/s}$. A comet that slams into the Sun, which occasionally happens, would have a speed close to this maximum. Alas, it's also the last speed it'll have before it meets its doom :-)
Update
If you want to know the fastest object in the solar system that didn't crash into the Sun, then the best candidates are sungrazing comets, i.e. comets with very eccentric orbits that pass very close to the Sun. One particular group are the Kreutz Sungrazers. The comet C/2011 W3 (Lovejoy) mentioned by hobbs in the comments belongs to this group, but there was another of these comets that passed the Sun even closer: the Great Comet of 1843.
This comet has a perihelion of only 0.005460 AU (where 1 Astronomical Unit is 149 597 871 km). This means it came to within less than 121 000 km of the surface of the Sun, and amazingly it survived (most comets break up when they come this close). So what is its velocity at perihelion?
The general formula is (see this link) $$ v_p = \sqrt{\frac{\mu}{a}\frac{1+e}{1-e}}, $$ with $$ a = \frac{r_p + r_a}{2} $$ the semi-major axis, $r_p$ and $r_a$ the peri- and aphelion, $$ e = \frac{r_a-r_p}{r_a+r_p} $$ the eccentricity, and $\mu = GM_\odot$ the standard gravitational parameter of the Sun. So we can rewrite this as $$ v_p = \sqrt{\frac{2GM_\odot}{r_p}\left(\frac{r_a}{r_a+r_p}\right)}. $$ As you can see, this reduces indeed to the formula for the escape velocity if $r_a$ goes to infinity. For our comet, $r_p = 0.005460$ AU and $r_a = 156$ AU, and we find $$ v_p = 570\;\text{km/s}. $$
When there aren't comets falling into the sun, Mercury is hard to beat. This NASA fact sheet lists Mercury's orbital velocity around the sun as varying from $38.86$ to $58.98$ km/sec, not so much greater than Earth (less than a factor $2$, even at maximum).
A comet doesn't need to impact the sun in order to come very close to solar escape velocity at perihelion. There is a class of comets known as sungrazers that pass very close to the sun. Although small ones evaporate on their first pass near the sun, larger ones can survive several orbits, and be considered periodic comets.
There is a class of sungrazing comets called the Kreutz family that has a very low perihelion and a reasonably high aphelion (150-200 AU) making them the best candidates that I know of for "fastest object in the solar system" when they pass near the sun. The comet Lovejoy (C/2011 W3) has an aphelion around 157 AU and a perihelion of 0.00555 AU (within the solar corona, note that the sun itself has a photosphere radius of 0.00465 AU!). As such it passed by the sun in December 2011 at a speed of 536 km/s, within a couple percent of the escape velocity at that height, which is 565 km/s. The Great Comet of 1843, another Kreutz-family comet, reportedly passed even lower without disintegrating, 0.00546 AU, giving it a speed of 570 km/s.
Pulsar did a fine job of working out the math, so I won't duplicate it here, except to emphasize the point that once your aphelion is tens of thousands of times higher than your perihelion, aphelion stops making much of a difference. If you're 100km above the surface of the sun and travelling at hundreds of km/s, the difference between the speed you need to go to get 100 AU out, and the speed you need to go to get 1000 AU out, is miniscule, and both are very close to escape velocity.