Perhaps the most consistent way to add smoke flavor using a gas grill is to purchase an Amazen smoker tube. No moving parts, and uses pellets or small old chips - tho I find the uniformity of pellets to be better.
Look at the Site Sponsor section for Amazen smoker.
Now, given the correct circumstances - you can "pre-smoke" or "cold smoke" what you plan on grilling and gain the cooked over wood flavor.
But, you want to be careful - in that ground meat and poultry are particularly prone to propagation of salmonella if not handled correctly. Look at the food safety section for some ideas. Whole muscle cuts of meat vs ground are better candidates for pre cook smoking especially during the warmer time of the year - but again, read, read, read to get comfortable with food safety. No amount of advice will keep you safe from yourself
....So, also reference the USDA website for safe food handling practices. Lots to learn there.
Otherwise - you, my friend - need a smoker
. And - there are SO many choices - I don't want to give you a headache. Step at a time!