You must apply and get accepted at the school the normal application process. However, you must let the school know that you are a LANL employee applying and planning on doing your graduate work at LANL.
You must be an accepted student at the school you wish to take classes from, and your line management must approve of the time you spend during the work day on your educational activities. The amount of time that a group will approve varies.
This will depend on what your line management approves. There is fundamentally no issue with taking some courses on campus and some here at LANL. You can do all work here at LANL, but you may only be able to take one or two courses per quarter/semester.
LANL employees are considered "in-state" at the University of California schools. The estimated costs for attending can be viewed at the University's website. There is an additional fee that must be paid to LANL if you intend to take the class from here; the exact prices vary slightly by school and are posted on the course registration page.
Costs do fall under the tuition assistance/reimbursement program. It is likely that you will have to address up-front payment with you management. Tuition reimbursement is contingent on getting a satisfactory grade in the class (B or better for graduate level classes). That would imply that it may be difficult to get payment up-front.