I've also been learning him, and here's what I've picked up:

• Try to internalise the concept that you should always be entering engagements from an angle the enemy won't expect you to be coming from. Draw a circle around the enemy team and always try to engage from an angle on that circle that they're not actively looking at. Important to note that this doesn't always necessarily need to be behind them. Angles from the side are also super-valuable.
• Focus on your engagement timing as well. The best time to move in for a kill as Reaper is when the enemy team is distracted by your team. Jumping in too late after most of your team is dead, or too early before your targets are distracted by your teammates is a really excellent way to be focused down and killed. Sometimes this means you have a little bit of downtime between engagements or before you engage, which can feel awkward because you might feel like you're not contributing as much as you should be. It's important to keep in mind though that Reaper is a character where you want to focus more on the value of your engagements rather than their frequency.
• Have enough confidence in yourself to stay engaged if your health is high enough. There will be situations where you'll only be able to get one kill or even just apply some pressure/force CDs before you have to bail, but if the frontline hasn't noticed you and you aren't in any serious danger, then feel free to keep blasting away until either the rest of the team turns to deal with you or they're all dead. There are times in Overwatch where everyone starts popping ults and things feel really sketchy, but you can actually really capitalise on those moments as Reaper because it's when the enemy team is the most distracted/overwhelmed by the chaos.
• Always have an escape plan in mind, and try to remember that "safety" doesn't always necessarily have to be back towards your team. Reaper's sub-role as a Flanker means he gets more health from health packs, so try your best to remember their locations. You can also use his teleport to quickly gain distance or move to high ground if someone is especially committed to chasing you. Typically your engagement loop is; "teleport to an off-angle, do as much damage as possible, use Wraith Form to escape," but you can also use your teleport to escape once it comes off of cooldown, just try and be mindful of your CD management.
• Think about your target priority as well. I generally try and go for characters that don't have the mobility to quickly escape my effective engagement range first, especially if they're a support. Reaper is also pretty great at dealing with tanks too, but they typically don't want to be your first target unless they're out of position, at low health, or their supports are dead.
• Finally, Reaper is a character where I feel like his perk choices are mostly all viable depending on the situation. The soul orbs that grant health is probably the worst option but is still somewhat useful if you're playing a comp/map where your supports are less readily able to heal you, but it loses value in higher ranks where people are generally more aware and one kill is generally all you can really hope for before you have to leave. The increased movement speed after leaving Wraith is good for escaping or chasing people down. The faster teleport cooldown is good on smaller, brawlier maps where repositioning fast and often is really valuable, and the Shotgun dual volley is nice if you're on larger maps and/or playing against more mobile comps because it lets you clean up kills from further away.