If the process is killed partway through, then you either ran out of memory or you ran out of disk space. I'm going to assume disk space. When compressing your entire disk data, assume you'll need the same amount of free disk space left as the total number of bytes all of your data is that is to be compressed.
Your best bet if you want compression is to compress in pieces, but if you're transferring from server to server and you have enough bandwidth available (so that you're not billed extra), then do a direct disk copy from server to server so less strain is put on both systems, which means your old running server will not slow down as much.
Compression will slow things down somewhat because the server needs to make calculations on the data as well as manipulate it in order to compress it and thousands of calculations and manipulations take time. The advantage to compression though is less data is being sent on the wire which may mean lower bandwidth costs.