Depends on what type of throwline you're using. For slickline the bucket is great, just flake it (not coil it) into the bucket. If you don't want to carry a bucket into the woods then wrap the Slicklne in a figure eight coil between your elbow and thumb, this is the technique Treeman teaches in the BTCC. When you get to the tree flake the line from the coil onto the ground or a small tarp or your shirt to throw.
For the skinny throwlines you always want to flake them into your container. The bigger the opening on the container the easier it is to flake the line in and the easier it comes out when you throw. New Tribe sells a big yellow bag that works great for the skinny throwlines:
Big Line Mug
and you can also stuff Slickline into it but you'll need to flake it on the ground to throw
The ultimate in luxury and ease for the skinny lines is the Falthheimer cube or the Sherrill domestic version. This is a durable folding cube that allows you to have a throwbag pre-attached to each end of a flaked throwline. It folds up small, easy to carry. When you get to the tree just open and throw. It's very quick to reload the cube on missed throws and is really helpful when you're in the woods with all kinds of sticks and shrubs etc. on the ground. Well worth the money, look in this category on Sherrill
Throwline, Big Shot & Weights > Throwline Storage
-moss
