In something of an inverse of what you might think, warriors have a hell of a time fighting dragons. If you’re playing a pure melee character without the benefit of spells or bows then you’ll have only your Shout attacks to battle the behemoths. As you might guess it’s not exactly peaches and rainbows when that dragon really wants to roast you alive from the sky without actually ever landing to battle you. That leaves you with one option, wait for it to land and…
Above: Get right up in its face
A melee character is best off avoiding dealing with the dragons as much as he can until acquiring the Dragonrend shout. When used this reduces a dragon's defenses while also forcing it to land. This Shout is obtained by following the main storyline until completion of the quest Alduin’s Bane. Upon acquiring this Thu’um you can force the dragon to never even take off, which allows you to just run up to it with your weapon of choice to beat its face in.
The above may sound simple but until you get that ability every fight with a dragon is going to be a trial of patience if you don’t want to use a bow or magic. Your best bet is to have a follower who does use magic or a bow and then put yourself under cover. Step inside of a tent, under an awning or just out of view with your shield raised (if you have the perk Elemental Protection). Once the dragon lands, charge in and beat on it, preferably before it hurts you too much. Rinse and repeat to end the fight.
You may find yourself using your Shouts a lot to get ranged damage in. If this is the case try to get an Amulet of Talos as it reduces the cooldown on your Shouts by 20% which is a huge boon.
A pure melee thief setup isn’t going to have a whole lot of success dealing with dragons. The problem is that you can’t use sneak attacks on them and you’re probably using daggers, which don’t deal damage fast enough to kill dragons with any expedience. For these situations you’re going to have to use a powerful sword to make up the difference. The best thing to do is use a bow for dragon, mammoth and giant battles to gain experience enough with it to do serious damage to dragons with your arrows. Equip the strongest bow you can get, give it a good enchantment and then buy the best arrows available to deal maximum damage.
Even if you are sticking to using your bow for all encounters you might want to bring a shield to any dragon battles. Stick a Fortify Health enchantment onto the shield to enable yourself to stand toe to toe with the beast if it gets right in your face. However you’ll want to pull back constantly to try to get some space, then use the quick change menu to swap back to the bow. The video below is a pure thief character who has gained a huge amount of one-handed skill thanks to the dragon fighting involved in creating this guide.
Above: Late games fight using the Dragonrend to make the battle easier
Honestly a thief build should have some of the easiest dragon fights out there. The bow is an incredibly powerful weapon when trained up a bit and it can hit the dragon no matter where it hangs out. You might think a warrior wading into melee would have the easiest time but until you get Dragonrend that dragon may simply choose to never land. If that happens the archer can still yank down their health rapidly while a melee build stands there looking silly.
There are two distinct types of mages: the ones who run around in robes and use pure magic and the ones who use armor and weapons to complement their magical abilities. Both of them have a pretty easy time dealing with dragons so long as they’re given the right abilities to handle each dragon that comes after them.
The pure mage has a tough time since for the most part only destruction magic is helpful against dragons although summoning up an Atronach can help while the enemy is at a distance. Illusion just flat out doesn’t work without high level perks and Alteration is mostly useless. Having a decently leveled Destruction skill with the right perks will allow you to hammer on the dragons when they’re both at a distance and close by.
Above: Burn the dragon!
A good strategy to deal with dragons is as follows, using an ice dragon as the example: Summon up a Flame Atronach and start pummeling the dragon with fireballs (one-handed to preserve magicka) until it moves closer. Once it does so dual-wield the Flames spell to burn the living crap out of the beast, guzzling down potions to keep your magicka up until the beast either dies or flees back up into the air. Note that since lightning spells fire instantly, they're great for hitting flying dragons since you don't have to lead your shots, and they work for getting a dragon to land, too.
One of the safest ways to kill dragons with a mage character is if you’re using a battle mage type of character. Slap on a strong suit of armor that you’ve enchanted to enhance your survivability (resistances or health increase) then get a weapon that will drain HP from your enemies - a sword works best as it strikes fastest. Circle strafe around the dragon using your spell of choice while hitting the dragon with your sword. The dragon will often get confused and keep turning to keep up with you while you’re pulling his health down and keeping your own up.
Above: Something like this
Frankly a mage has an easy enough time so long as you can get a bunch of equipment on to bolster both your base magicka levels and regeneration rate.