Performing large amounts of schema updates and transaction log issues
Starting with SQL Server 2012 adding of not NULL column with default values to an existing table is metadata only operation.

From what you wrote it's clear that it's not your case, i.e. or your server is beyond 2012, or one of the following probably is your case:


the newly added column increases the maximum possible row size over
the 8060 bytes limit the column cannot be added online
your server Edition is not Enterprise


(you can find the complete list of restictions here: Online non-NULL with values column add in SQL Server 2012

So how you can resolve the issue?
If you cannot upgrade to at least 2012 or your Edition is not Enterprise, the only thing I can suggest you is to split this one big transaction to many transactions. This will permit you to clear the log (on checkpoint in simple recovery model or on a log backup in full / bulk logged).

So instead of adding not null column first add null column and then update it with your values in batches. So the log will have the possibility to clear between batches.