1) Finishing each battle (the ones that lock you into a particular area until all enemies are defeated) within the time alloted is key. Some are harder than others. In case some of you haven't noticed, during each battle there is a small timer counting down you can see under your score, al-la Ninja Gaiden.
2) Dash through as much of the level as possible to get a better overall TIME BONUS score.
3) Take as little damage as possible, and try not to die.
4)
Dash + Slash is your friend. It packs some power for a single hit, and will decapitate and of the normal sized enemies (like the Vampire Males and the Bladed Arm guys that slide along the floor). Correctly utilizing this move can "cut" down on the time it takes to finish off your enemies.
5) Those Fat Blood Filled Guys can be grabbed after they are stunned. The easiest way to do this is to use the
Dash + Slash. You will see them stagger back a few feet. If you're close enough in front of them while staggering you can grab them. To get the time bonus for certain battles, utilizing this to kill them faster through explosion can save a lot of time.
6) The time bonus at the end is fairly forgiving, allowing you to get most of the treasures along you path. This really helps to boost your score.
7) Use your gun when you have to. Once you've collected enough red sculls, you'll have enough bullets to play around with. USE THEM for quicker kills!

I've found that to get the time bonus for some battles, you really need to utilize to explosion correctly. Say you have 5 or 6 guys lined up in a row. From one side of the screen, shoot the majority of them down with a single shot, then slash + grab the guy closest to you then explode him. If they're close enough, all the remaining enemies will blow up as well.
9) Those annoying as hell frog things (the ones that electrocute you when they touch you) can be killed in a single hit with
Dash + Slash.