Shaming is counterproductive; here's an article on shaming as a bad motivator to losing weight. So, positive rewards for good behavior are the way to go.
Three Goals To This Game
That leaves you with a few goals and many strategies. For goals, you want 1) top performers to push the ceiling up and set new standards, 2) worst performers to do better and move the floor up, and 3) the center mass to move into better performance.
The priority of these goals depends on their impact. Do top performers outweigh bottom performers? Do bottom performers drag everyone down disproportionately? Whichever will have the most impact as a bottleneck, make that your top priority, but keep the other two as your next priorities. Eli Goldratt talks about this with his Theory of Contstraints, and in his book The Goal.
In the book Gamification by Design (pg. 20), Zichermann & Cunningham identify four motivations for gameplay: for mastery, to de-stress, to have fun, and to socialize. Consider these elements and how you might use them to motivate.
Badges and points are a good start. Badges give good qualitative, feel-good rewards. In my experience, they incite "collector" behavior. You will need to come out with bigger and bigger badges, but can set the goals on a geometric scale (rather than linear), so that the difficulty in leveling up is commensurate with the skills people have gained. The administrative commitment for badges is time to come up with new badges.
Point Systems and Rewards offer more tangible incentives. Give redeemable points for certain behaviors and create a "redemption catalog" of things people can use their points toward. Gift cards are a typical incentive. Redeemable points accrue until they are spent, or "burned" off. Give experience points for overall levels of achievement, and mabye bigger prizes. Experience points only grow. (Zichermann & Cunningham pp. 38–39)
In order to motivate your worst players, put bottom performers on teams with higher performers. Obviously, these employees may have no social or functional connections whatsoever, so you'll want to engineer a virtual connection within the game. Teams get points for overall performance, so better performers have an incentive to help out their less-skilled peers. (It's just like group projects in school where you get points for individual contribution and points for the overall group output).
As for other specific strategies, look at using different types of leaderboards. The overall leaderboard is only interesting if you're at the top. Local optima give people achievable goals, so go for monthly and weekly leaders. That way, recent improvement becomes just as visible. (And g)Give badges for staying on the top of a leaderboard for a run.)
Cash Rewards are a straightforward approach borrowed from the loyalty program industry (Safeway and Tesco cards, Capital One Rewards, that sort of thing). Consider cash rewards given for advancing in any of the metrics above (badges, experience, team performance).
Metadata in a dashboard or on profiles can add depth and more options for personal bests. Since this game is about time reporting, you might include data showing how people's performance contributes to the bottom line. If billable rates are known internally (and not secret to protect salary information), use them to calculate real dollar value of performance. Consider the interest earned on billed hours that comes in 30 days sooner than otherwise. You might also show accuracy by percentage for each person, and even use these as a qualifier for points earned (with 100% accuracy, you get more points than with 85% accuracy).
Just the Beginning
There are plenty of other strategies and tactics, but this should give you more than enough to get started.