How can I insert into a Destination Table from multiple tables concurrently..?
This article is for you then. 

One quote directly in regards to your question of multi-threading.


  The package property that determines how many executables can run in parallel is MaxConcurrentExecutables, as shown in Figure 2. The default is -1, which means that the control flow engine will use the number of logical processors on the server plus 2. For example, if you’re executing the package with the default setting on a quad-core server, you can have up to six executables running in parallel.


This article will talk about avoiding blocking and locking and should give you enough examples and ideas to give you your own questions to research on. Evaluating what types of locks and blocks will occur and what level of isolation you need will help you avoid blocking and locking. 

Deadlocking does not sound like a problem you will encounter based on what you have typed. Here's a good explanation on how it occurs.