How do I increase the amount of REM sleep I get at night?

Get full night sleep. It is the second half of the night that the percentage of time spent in rem sleep is greatest. The trick is to get a full night's sleep. Rem can be suppressed by alcohol, stimulants, and sedatives; avoid caffeine, evening alcohol, and sleeping pills (ask your doc first if prescribed). If waking up tired despite fill night sleep, talk with your doctor about this; you may benefit from a sleep study.
What is the goal? You cannot control the amount of rem sleep without medications. Normally there is no advantage in increasing rem selectively. If you are able to achieve enough sleep, you should be ok. If you feel that your are not rested after sleeping, or have some other issues, talk to a sleep specialist.