Hello my name is Mats. This is a just a suggestion for a build i use udyr for sololaning that works pretty good for me atleast.Its not especially advanced or anything at all. I hope u will find some good tips and guidance for how to build YOUR Udyr. I hope ya like ^^.

9 armor pen reds 9 Dodge yellow 9 in magic resist blue 1 attack speed and 2 Hp quintesesseseseseses.

Armor Pen: I use the Red Armor Pen as I am going Tiger stance for damage, a lot of players usually focus Udyr as hes a big threat in general.

Dodge: Goes dodge yellow for first of all udyrs passive, which does great sustain in lane.

Magic Resist: Straight forward defence against AP based champs. Will do a big contribute in surviving a lane against a singed for instance, or a ganking nunu/mumu.

Entering a stance grants Udyr 10% attack speed and 3% dodge for 5 seconds. This effect can stack.

A decent passive that becomes more important as the game goes on and you have mana to maintain stacks. It caps at 3 stacks totaling 30% attack speed and 9% dodge bonus.

Persistent Effect: Udyr's attack speed is increased by 20/25/30/35/40%.

Activation: Udyr gains an additional 15/20/25/30/35% attack speed for 5 seconds after activating Tiger stance. While active or for 5 second after Activation, Udyr's next attack deals 170% his attack damage plus 30/80/130/180/230 over 2 seconds as magic damage.

Persistent Effect: Udyr's attacks cannot crit, but he restores 12/14/16/18/20% of his damage as health and 9/10.5/12/13.5/15% of his damage as mana.

Activation: Udyr gains a temporary shield that absorbs 60/100/140/180/220 (+0.5 AP) damage.

Persistent Effect: Udyr mauls his target with such force that they are stunned for 1 second. This effect cannot occur on the same target for 6 seconds.

Activation: Udyr gains 15/18/21/24/27% increased movement speed for 2/2.5/3/3.5/4 seconds.

Persistent Effect: Every fourth attack, Udyr engulfs enemies in front of him in flame, dealing 50/90/130/170/210(+0.25 AP) magic damage.In addition, Udyr's ability power is increased by 16/24/32/40/48 and his attack damage is increased by half of that amount.

Activation: Udyr sends out pulsing waves of fire dealing 15/25/35/45/55 (+0.25 AP) magic damage each second to nearby enemies for 5 seconds.

For Masteries i run 0/11/19 simply for Tankiness and sustain, because i believe Udyrs q ability speaks for itself when it comes to dmg. in defense i make sure to take 4 in evasion for dodge stats + the Nimbleness (movement for udyr is key). Why i go 19 in utility is simply for (movement speed), CDR, neutral buff duration ANNNNDDD increased stats for the Ghost and Flash summoner spells which i always run as Lanedyr.

Dodge boots: For enchanted movement 2 (lolz) and increased dodge stats.
Wriggles Lantern: It gives me great lifesteal= survivability, it helps me push the lane faster, it gives me increased dmg with q and its a free ward.
Trinity Force: gives me increased movement which is very much needed, plus overall stats really.
Spirit Visage: Gives me increased cooldown so i can spam my stances faster, but mostly for survivability.
Fource Of Nature or Frozen heart: depends on if i want to just move faster and have increased magic resist or if i want armor tankiness and Cdr to spam spells faster(passive yo combocombo).
GA: if its one of those long intense can-go-either-way games when the enemy carries gets all their items and ur tankiness isnt as extreme tanky anymore ( udyr is a big threat in teamfight and can get focused), its always good with an GA since udyr in teamfights usally runs into the fray to find the right target to take out.

I start with bear stance (E) at lvl 1 and just leaves it there. Rest i maximixe first q (tiger) and then w (turtle) because q deals bat**** crazy dmg, liek 180 at lvl 1, and turtle keeps me alive in lane, and if incoming ganks.

i Take flash and ghost as summoner spells. simply to help catch up to enemies. And also to quickly initiate or get out of teamfights/incoming ganks etc..