Yo can add "Execute Shell" build step for your Jenkins project with shell commands to run you JMeter Test Plan without UI. Something like this:
sh $jmeterdir/jmeter.sh -t $jmeterdir/HealthCheck.jmx -n -j jmeter.log
In order to check results of JMeter Test Plan executing for making a decision about Build Failure/Success you can implement the following approach:
1. Add Simple Data Writer to your Test Plan and make it log errors only into some file
2. Create sh script which takes you error log as parameter and check number of lines in it. If number of lines not equal to 1 - it means that we have at least one error in error log - do "exit 1" - this fails build:
count=$(wc -l < BuildErrors.csv)
if test $count != "1"
then
echo "There are errors during test."
exit 1
3. Run this script from Jenkins "Execute Shell" build step