When I started running I thought I was pretty good at going up. I was able to maintain my pace and pass other people in road races that had inclines and I used to pride myself on running every step in training. It was when I became more adventurous, entered some proper mountain races and experienced being mobbed by the entire field that I realised I actually sucked at it. What I'd thought was a strength was actually a weakness and if I wanted to be competitive I needed to improve. In previous races it was just a decent level of general fitness that had got me through any lumpy bits, but it takes more than that to tackle proper climbs effectively. I spent a lot of time experimenting and my performance and efficiency reaped the rewards.
Practice and exposure are the main things needed to get better at running uphill, but there are techniques that allow you to use less energy. There is also a point where it is more economical to walk. If you can maintain a decent hiking pace then it's almost as quick as running, plus you give your 'running muscles' a break and you're fresher when you reach the top. This is noticeable in races; I've often switched to hiking and someone has overtaken me absolutely blowing out of their backside but feeling delighted to have passed the weak opponent who was forced to walk, only for them to reach the top heaving and purple in the face allowing me to hoon it past them on the subsequent plateau or descent. Ignore what other people are doing; it's not a failure to walk. Call it hiking or speed-hiking if you prefer!
Speed-hiking a.k.a. walking
Reknowned coach Jason Koop wrote a good article digging deeper into when to walk and when to run, which you can find here
You can always choose an inbetween option too and adopt a run/walk strategy. I first tried this in the 100k de Millau, which is an undulating road race that has four particularly killer hills. The inclines were just about runnable, but I felt I was going too far into the red for that length of race, so I started counting my steps and I ran 20, hiked 20. I often found that I wanted to keep running for longer than 20 steps, so I allowed myself to run further if I wanted to, but maintained 20 steps for the walk breaks. It really makes a climb more manageable but it stops you from slacking off too. I've used it in every hilly race I've done since and it's amazing how easily you can actually get running again if you know you're 'allowed' to have a walk recovery after a few steps.
The 100k de Millau 2016. I finished in 9h26, was 2nd female and 26th overall out of 1275 finishers. Full results here I entered this race on a whim straight after doing the TDS because I was itching to stretch my legs out and properly run rather than hike. My friend Marion supported me on her bike; I don't think many runners were fortunate enough to have a cyclist of her calibre accompanying them! See her in action here
If the incline is shallow enough for you to run then you can tweak your technique to become more efficient and to rely less on the strength of your muscles. As well as through experimentation I got ideas from YouTube videos and from the excellent Chi Running book. This is what works for me:
Take shorter strides, using tiny baby steps if necessary.
Increase your cadence. It may go as high as 200 steps per minute or more.
Relax your neck and shoulders; look for a feeling of lengthening in the back of the neck and allow your head to lead you upwards.
Consciously focus on muscles other than your quads. Engage the glutes, engage the core. Simply thinking about these muscles is usually enough to call them into action. Your quads are big muscles; they need a lot of energy and tire easily. Spread the load and let your posterior chain (glutes especially) push you up rather than have your quads pull you up. Experiment with finding the feeling that your leg muscles are quite relaxed and that it is your core that is doing most of the work. Again, simply thinking about these muscles makes you more conscious of them and will help them engage.
Use your arms and upper body to help propel you; if you're unaware of how much you use your arms then try going uphill with your arms clamped to your sides.
Lean slightly into the hill.
If you switch to walking/hiking then the principles are the same, although you may choose to take bigger steps and have a lower cadence. This is a very personal thing and some people (me included) prefer to take fewer steps, gaining more height with each one. If you go up flights of stairs taking two at a time chances are taking fewer bigger steps will suit you!
If you're training for a race and are likely to hike during the event then make sure you practice it in training. There's a big difference between settling into walking along in your comfort zone and hiking purposefully.
The use of hiking poles is also very subjective. There is no doubt that they help spread the load between lower and upper body and these days most elite athletes use them in races such as the UTMB. Purists think of them as 'cheating sticks' but they're allowed in most races so if you find they help you then don't hestitate. Personally I occasionally use them in very long, mountainous ultras (let's say over 80k/50 miles), but I find them cumbersome on the descents and I prefer to run without them.
Using cheating sticks in the TDS 2016. That race had an individual climb of 2000m elevation gain, which came at 50k into the 120k race, so I felt poles were justified!