Storm is what is called a glass cannon in experienced gamer circles, a character that can do enormous damage but is quite fragile. Storm has the lowest health of any school, and resistance provided by storm gear is also usually lower than other schools, so running out of health early in a duel is a real problem. I am running a level 71 storm in Zafaria at the moment and have some suggestions, most of which probably aren't news to you but maybe one or two will help:
1) Gear. The universal resist provided by waterworks gear is a godsend. It may not seem like a ton, the total resist for hat, robe, and boots together give 26% uni resist, but I have to say that 26% makes an enormous difference in surviving fights. Farming WW is a pain, but there is a reason people do it, and storm probably benefits more from WW gear than most schools. If you can't get waterworks gear (Skyscream hood, cape and boots) at least go the the bazaar and buy some of the better bazaar gear. There are auctionable gear items available at the bazaar that provide quite good resistance vs elemental attacks, such as the Tempest Topper and Eureka Wraps. Resist vs spirit or balance attacks is harder to come by, unfortunately. There is craftable Wintertusk gear that provides universal resist, but the resist numbers are lower than waterworks gear.
2) Pet. A good pet with "may cast" healing spells and Spell Proof is almost as helpful as WW gear. No first generation pet is really good for this purpose, you do much better getting someone with an awesome pet bred for optimized defenses to hatch with you. Fortunately, there is a group that specializes in this, Pet Central. Visit their thread on the Wizard101central fansite forum and set up a "pet mixing meetup". You'll need to set up a username on the fansite, but that's free and only takes a minute. These folks provide a highly useful service at no gain to themselves, so please be polite and patient with them.
3) Defensive spells. Build your deck with an eye to defense. Some of the more useful defensive spells will need to be bought with training points, so be sure to spend your TPs wisely.
Storm needs healing spells to survive. Forget about Healing Current, it's widely varying results can't be counted on. You are going to need life spells. Training life to Satyr for 7 training points is the favored option here. If you don't have the training points for that you may be able to get by with treasure cards. As of this writing I see Satyr TCs available on the bazaar for 550 gold, Regenerate TCs for 2400, and Dryad TCs for 1350. Satyr TCs can be bought from the Zafaria librarian for 1,000 gold, but are usually cheaper at the bazaar. To get the most from these spells you are going to need a life mastery amulet.
Healing spells aren't enough, not for fragile storm. You need shields. OK, you can maybe get by without them if you always fight with a buddy, or in street fights with non-boss mobs. But if you want to be able to solo the tough bosses, shields are a must. Think of every shield you cast as giving you extra health. If you get hit by a 500 point attack and have a 70% shield in place, it stops 350 damage, and effectively just gave you 350 health. Not bad for a zero pip spell. Many players advocate training ice to Tower, and using Tower Shields, but you actually do better if you use more specific shields. As storm you get two useful shield spells from your school, Thermic Shield covers fire and ice with 70% resist wards, and Storm Shield covers storm with an 80% ward. If you have trained life to Satyr you get Legend Shield, which covers death and myth damage. To cover life, you can visit Sabrina Greenstar in the commons and train Life Shield for 1 TP, or if you have trained death spells you can use Dream Shield. Before each boss fight check the boss's and minion's damage types and add appropriate shields to your deck, and remove shields that won't be useful.
4) Offensive spells. It is said that the best defense is a good offense, and this definitely true of storm wizards. Defeating your opponents quickly means having to survive their attacks for fewer rounds. Good deck building is key, you want your deck to have everything you need in it, but nothing you don't, every useless spell in your deck makes it harder to draw the crucial spells you need. Nothing is more likely to lead to defeat than to have to sit through several rounds taking pass options because you can't draw the card you need.
You want attacks, lots of blades, a few traps, sun damage enchantments, a few healing spells, and maybe half a dozen shields. I usually enter boss fights with 3 Stormlords and 2 Tritons; I use a Stormlord first to wipe out the minion(s), then finish off the boss with a Triton. Against tougher minions or bosses I may substitute Sirens for Stormlord or Leviathan for Triton. The extra attack cards are there to let me draw the one I need faster and as insurance against fizzles, I rarely use more than one of each. A single Tempest makes a good finisher if I mess up and leave someone alive with a few hundred health. I use Gargantua as my sun damage enchantment, when I reach Drum Jungle I will train Colossal and use it instead. For blades you of course want several Storm Blades. Elemental Blade is highly recommended, you can learn it from Niles on Krokosphinx island for 1 TP, bring two. I usually slip in a Supercharge or two, but don't always use them. Be sure to include at least one Darkwind, I usually include two to increase the odds of drawing one early. A Storm Trap or two is worthwhile. If you have trained death to Feint, put in a couple of Feints. I consider Windstorm and Elemental Trap as rather poor value for the pip they cost, at most put in one of each. Then round out your deck with shields and healing spells as discussed in 3) above. I typically put two Satyrs and six shields in my deck, more against harder bosses.
I am looking for the best setup for PvE combat that will help me do solo only bosses and not die within four rounds of battling and recommended cards and deck would be nice also.
My storm does have to treat solo boss fights a lot different the the wham bam done with regular questing. I usually start by looking to see if there is a guide for it on Wizard101central.com, many are old but can give you the general idea of how to structure the fight and what to expect while in it. I usually bring in the shields for these fights, something I am not great at. Either reshuffle or TC back ups if I run low on cards since I use a minimal deck set up, storm is least able to wait for a spell to come up. For some you could even switch out gear with more resist to their school. A good known to heal pet and I am usually good to go. My storm passed on extra healing spells as I do not have the amulet so I have the original 2 pip one. I also can get enough damage out of spells that I don't rely on Feint and did not train it. The monsters put enough traps out adding another to myself seems a bit dangerous. I have taken all spells and craft gear that helps boost my damage and give me extra blades/traps, such at the helmet crafted from Wintertusk. Having several different blades and traps is a big help. Shield and maybe shield again, blade, trap, fire out a wave or storm lord? Finish up boss with one hit if he's still standing.
I am currently at the end 2 dungeons in CL with this wizard and am trying to solo as far as I can in the game. Take defeat in stride and learn from it. Glass Cannon is the right definition of storm.