Hum.... it seems it might be somewhat difficult to fully strip, but you can try getting some brake parts(non-chlorinated)/carb cleaner and using the straw to spray out nooks and crannies. It'll be best to take it down as much as you can first. Here's a diagram of how the parts go together (I think.... a savage 110E is the same as stevens 110E, right?)
http://stevespages.com/jpg/ipb-savage-110c-d-e-k.jpg
I'd say, spray everything out with cleaner and then get some breakfree CLP and let that stuff soak in for a while... then load up with a snap-cap so you can dry-fire it a bit to work the mechanism before trying with a live round again. And make sure you clean out the barrel before you do that, too!
Quite a few actions are made to be taken apart with non-specialized tools... as long as you're mechanically handy you shouldn't be in too much of a pickle if you can just take things out, clean em up, and put em back like yeh found em. Be careful of any pins that don't drift out easily as they may be under tension from a spring.
Start slow, clean a few pieces, then put em back together. The next time you take it down, go a few more pieces into it, and soon you'll know the internal workings like the back o yer hand.