It seems like I'm having to constantly remove items from my inventory because the maximum weight of 300 keep getting surpassed, thus preventing me from running. Is there any way to increase the maximum weight that I am able to carry?

1) put more points into stamina ((not recommended if you are a mage))
2) Find the "Extra Pockets" perk underneath either lockpick or sneak to get 100 added weight
3) Find the enchantment that has carry weight and then enchant stuff to have that enchantment.

Well, you can up your stamina. Or you can enchant your armor to let your carry more.

Get the steed guardian stone, it's to the west of solitude.

Its useful buying a house to store the extra stuff you don't need right away.

Also get a horse. You don't have to ride it everywhere but when you are about to enter a dungeon, fast travel to the same location which will summon it to you. Even when overencumbered you can ride a horse at its normal speed and fast travel, so its useful to have around when you leave a dungeon with your haul.

If you are overencumbered in a dungeon use power attacks to move faster.

And you can boost your encumbrance limit with magical items (watch out for Boots of the Ox and similar in merchant inventories) and potions of strength and fortify encumbrance which you can brew yourself with alchemy (use creep clusters + somethine else). Your encumbrance limit can be increased naturally by increasing your stamina as well.

also your companion pretty much doubles what you can carry as you can use them as portable storage

Even though it doesn't specifically say so, putting points into stamina raises your carrying capacity. I'm at over 400 currently. As others have said, you can also find the steed stone or find gear enchanted with increased carrying capacity.

Up your stamina, enchant gear, the steed stone hundred extra pounds, a perk in pick pocketing that allows a hundred extra pounds.

Buy a house and drop off your dragon bones and scales, alchemical ingredients, food stuffs, unwanted tat you may have picked up by accident (countless pots and baskets) potions you'll never use (fortify magica for a warrior etc) stash books on shelves, spare weapons on racks.

It all mounts up, doing that and you'll suddenly have an extra 50-150 extra encumbrance.

There is this stone place where you can get blessing for an extra 100 carrying capacity and something else I forgot and is on the path leading up to Solitude to left or generally it is West of Solitude.

There are a few tricks outside the above mentioned ones:

1. Place all loot on a corpse and dead thrall him. Kill and rerise it whenever you need to add more stuff. Can also be done with the temporary versions as long as you have bodies in the exterior world so you can immediately fast travel. Save often because dead thralls have been known to disappear.

2. Place items in a container and ask your follower to take all items. This allows them to exceed their weight limit, although sometimes the items are flagged as stolen due to bug as it is not advisable to do that for items you want to keep.

The extra pockets perk reminds me of a question that I've had.

With having the steed stone, companions for mules, and carry weight boosting enchantments, is extra pockets a waste of a perk?

What if u accidently fus ro dah your companion into a waterfall and they feck off doIwn the river never to be seen again? Or off a mountain? Or...or....I dunno but Id be carefull using them as pack mules

I would be more worried about their inability to avoid trap triggers and kill you.

Never a truer word said my friend.

Bad times.

Steed stone adds 100. You can top off at 600. It's also a good idea to carry an extra set of armor that is enchanted for weight. Also a ring and necklace. If you doing a mission use your regular armor, necklace and ring. When you get to the limit switch those things to the enchanted ones. Yes it adds extra weight you have to carry by carrying two sets of armor but if your usInt light armor the enchantments make up for it.

If you use heavy armor there is a perk to make set set of armor you are wearing weightless (you do have to choose two quite useless perks before that though).

Also ask yourself, what do you need to carry. If you get a house you can store your dragon bone and scales, rare items, ingots etc there, and then don't pick up useless or worthless things. If you pick up things to sell, then know your vendors (I have four in Whiterun, which I frequently visits).

But the weihtlimit can be quite annoying, especially if you are a collector.

the best thing to do is to raise your stamina every level up.also use the steed stone as other op suggested.keep ur eye out for boots that let you carry more,i have found 2 so far when plundering but also keep eye on vendors.

my weight is 485 WITHOUT the steed stone which will raise it to 585!!!

my boots give me extra 40 and i have mask which gives me another 20 so my basic weight with out perks/apperall is 425

Go to the Steed Stone and receive the Blessing; it gives 100 added stamina/carry weight, negates armor weight and causes heavy armor to not slow you down when running (along with it not draining stamina as fast)
Otherwise, put points into Stamina when you level up (this is hardly worth it unless you are a warrior as the added weight is +5 per level, which is really negligible unless you put a lot into it)
You can also get boots that fortify carry weight up to +40 I believe (These are pretty rare as I have only found them a couple of times).

this perk is pointless if you plan on using the steed stone because it makes heavy armor weigh nothing as well as the +100...

Up stamina
Add enchantments that increase weight
Get the 50 perk in Pickpocket that increases your weight by 100
Get the Steed Stone which gives you plus 100 and no armor weight.