How can I smith higher levels of weapons?

Improving Weapons/Armor in Skyrim is a function of 2 major factors:

• Your total skill level in smithing. (Base skill as modified by Fortify Smithing Enchantments and Potions.) Stash away the best Smithing Enchant you find for each relevant armor slot. You can go beyond Exquisite to Flawless with a skill total of 100.
• Having the Smithing perk that corresponds to the equipment you're trying to improve. With the correct perk, you can go beyond Exquisite to Flawless with a skill total of only 57.

Smithing perks will almost halve the skill required for each level of improvement you make on your items, so this can be a rather big deal. In general with the Steel Smithing perk and a level appropriate Smithing skill, you will often be able to improve Skyforged Steel weapons to be better than any weapon for which you don't have the related perk and with only a steel ingot. With 3 perks & a Smithing total of 100 you can reach the Armor Cap with Steel Plate. With 1 perk & a total of 116 you can Cap your armor with just Steel. Doing the same with light armors is also possible with Dragonscale (5 perks, 100 Smithing), or even Elven (2 perks & 144 Smithing).

With the skyrim-dawnguard expansion, the best weapons in the game are Dragonbone.

You'll notice from the last two paragraphs that the left side of the Smithing tree is very attractive no matter what armor type you prefer. Though there is a Return-the-Lexicon quest that starts in Riften that not only speeds your Smithing XP gains but also makes Dwarven armor (and it's related perk) remarkably attractive. In fact, with Ancient Knowledge Dwarven armor will beat out Ebony armor and be trivially behind Dragonplate Armor. Given how easy it is to reach the armor cap, you should probably be choosing your Smithing perks based on your intended weapon choice.

FWIW: I like to raise my base skill by making jewelery using Gems and Transmuted Iron Ore. The skill gain per iron ore is much better than iron daggers, boosts your Alteration skill, and this jewelery can then later be used to grind Enchanting. (for example "Fortify Smithing" is an obvious choice, though for both sell price and XP gain, Waterbreathing (or Muffle for shoes) is a high value petty enchant and Fortify Sneak maximizes your return on grand souls. I generally consider all intermediate souls to be weapon fuel..)