Urs

Normals
5B - Good range, frame advantage, cancels to super or DP on hit, or fireball on block. Extremely useful to chain into.
6B - Overhead. Hitbox is longer than it looks. Fast, but nothing combos off it.

5C - Large poke, cancelable. Good for pressure since it can be canceled.
6C - Overhead. Chargeable, becomes unblockable at full charge. Hits out of the air pretty well too.

5D - Wall bounces on Counter-Hit. Solid poke, good payoff for a CH, but doesn't cancel into anything directly.
3D - Knocks down, very good range. Good to use as part of a mixup at a distance.

j.B - Cross Up.

Taunt - Urs can hold Taunt to continue revving his weapon. If you finish the whole animation, Urs will enter Heat Up for free. Extremely slow though, so completely worthless.
Chains
j.A j.C

2A 2B 2A ...
2A 5A 2A ...
2A 5B
5A 6B
5B 6C
Can't be done in Heat Up!

Heat Up Only Chains

5B 2B
5B f.C
5B 5D

f.C 2C
Specials
236A/C - Bari Fire Shot
A fireball. Small, long startup, and has short range, but recovers pretty fast. Can be used as a pressure tool, to help control space, or in some combos.
A version is slightly faster but travels less distance. Mixup between both to make it difficult for them to Gatchi.
623A/C - Bari Fire Blade
Standard Dragon Punch type attack, completely invincible startup but very easy to punish on block. Can be canceled to either super for extra damage. Both versions can be tech-rolled.
C version hits more times and does more damage but also is even less safe. However, in some combos the A version will actually give more damage than the C version.
For the A version, both hits may be canceled into Super, but for C version only the first two can be.
214B/D - Speed Glide
A fairly slow charging attack. Hits many times, but does fairly little damage. Generally unsafe on block, but can be spaced right so it can't be punished.
D version goes farther and hits more but is even less safe.
Not a move you will generally use very much, but useful in a few situations.
236B/D - Sonic Tackle
A quick shoulder ram. VERY fast, covers a lot of distance, and does fairly decent damage. Has a lot of recovery though, and is unsafe on block. Extremely useful for punishing pokes or wiffed moves from a distance. Also needed for many combos.
Has a low profile and can travel under some attacks. It will go under Odile and Cedric's projectiles, but will not go under Marco or Urs's.
D version covers more distance and does more damage, but has even more recovery.
On Counter-Hit, B version Wall Splats and D version Wall Bounces. Both can easily be comboed after on CH.

Supers

236236P - Full Throttle Road Master
Forward charging super. Works like a DP, fully invincible startup, easy to punish on block. Fairly strong and covers a lot of distance, making it a good way to end basic chains.
236236K - Basilisk Burst
Short range attack. If it connects, finishes with a powerful strike that launches. VERY strong, great to combo into when you can, and can be comboed off of in some situations.
Not invincible at all though, and very punishable on wiff or block. Also very short range, so you must be careful not to wiff it when comboing.
Heat Up
While Urs is in Heat Up, all of his special moves are enhanced. Unfortunately, Urs has a very short Heat Up duration and it doesn't really help him much with mixups, so it is relegated to just doing big damage combos. He also loses the distinction between the light and heavy versions of his moves.
236P - Bari Fire Shot
Projectile starts up faster and travels in a diagonal direction. On hit, it launches and can be comboed off of.
623P - Bari Fire Blade
Considerably stronger and does a knockdown strike at the end. Very good for ending combos.
214K - Speed Glide
Behaves fairly similar but hits 7 times now. Still unsafe on block. Fairly useful for some combos but that's about it.
236K - Sonic Tackle
Faster, stronger, and always causes wall bounce. An essential tool for Urs, sets up for easy big damage combos and can be used to punish from full screen for big damage.
In addition, Urs gains a few new Chains in this mode, and loses the ability to do 5B 6C (see below).
Kara-Cancel
Normally, Urs cannot cancel his 2B or 5B into his Sonic Tackle. However, this can be done with a very tricky Kara-Cancel. Kara-Canceling is when you cancel the startup of a normal into a special move, and generally appears like the normal never came out at all because it was canceled so fast.
The method for doing this depends basically requires you to chain into another normal, but then cancel into the Sonic Tackle before that normal comes out.

For example, Urs can cancel 2B into 2A. So he can do:
2B 2B 2A-Kara Canceled into 236B
In this case, the 2A does not come out, but instead 2B appears to cancel directly into 236B.

The same can be done from 5B into 6C:
2B 2A 5B 6C-Kara Canceled into 236B
Which is considerably harder because of the input motion required.

In Heat Up mode, Urs can only cancel 5B into f.C or D. This is actually a good thing though, because it makes his Kara-Cancel off 5B considerably easier:
2B 2A 5B D-Kara Canceled into 236K

These techniques are quite difficult and require very strict timing. However, they give Urs additional combo options which can help him out. Learning them is important to mastering Urs, but one of the hardest things to get reliably in the game.
Combos

Basics

2B 2B 623C (236236P)

2B 2A 5B 623A
2B 2A 5B 236236P
On block, 5B can be canceled to a projectile or 6C instead. Very useful combo set.
623C(2) 236236P/K
Strong reversal damage. K super is stronger but shorter ranged and better VS air.
2B 2B 2A->236B
2B 2A 5B 6C->236B
Karacancel combos. Very useful, but very hard.
c.C 623C
c.C 623A(2) 236236K

c.C 236B, 236D
Works on Watson anywhere.
(Corner)
c.C 236B, B 623A
Works on Donvalve only.
CH 236B/D, 236D
CH 236D, 236236K

CH 5D, 236D or 236236K

Gatchi Counter

(1 Far)
GC 236B+D, 214D, 623C
[1322]

(0 Close)
GC c.C 623C
[1014]
(1 Close)
GC c.C 623A(2) 236236K
[1417]

(1 Close)
GC c.C, 236B+D, 214D, 623C
[1653]
Works on anyone, but harder on some than others. Link is somewhat tricky but well worth the damage increase.
(0 Close)
GC c.C 236A/C, 236D
[1030]
(1 Close)
GC c.C 236A/C, 236B+D, 214D, 623C
[1740]
Use 236C on Olivia, Odile, Ashley, Freed
Use 236A on Donvalve, Watson, Urs
Doesn't work on Marco, Deathbringer, Face, Coyori, Cedric
Works on Face in the Corner (236C)
(1 Close, Deep Corner)
CH/GC c.C 236C, 623C(2) 236236P
[1458]
Better for the corner.

High Gatchi Drive

(0 Any)
j.D |> j.C (late) |> 6C
[761]
Easy combo, but poor damage. Works decently for near the corner.
(0 Mid)
c.C, D, 236D
[849]

(1+ Any)
Heat Up, j.D |> 236C, 236K, 236A/214D, 6 c.C 236C, 66 623C or 236236K
[~1400/1702]
Use 236A at mid screen, or 214D if they are bouncing back from the corner quickly.

Low Gatchi Drive

66, j.B j.C |> c.C (f.C) 623C
[1116]
66, j.B j.C |> c.C (f.C) 623C(2) 236236K, 623A
[1518]
f.C might cause the DP to miss some characters.

Heat Up

2B 2A 5B 5D->236K, 214K, 623P
Kara Cancel combo in Heat-Up mode. Even if you mess up the Kara, it's a solid, safe combo!
236D, 214K, 623C

(Corner)
c.C 236P, 214K, c.C 623C
c.C 236P, 214K, 5B 623P
May need to skip the 5B on some characters.
c.C 236P, 236K, 214K, 623P
Stronger, but much harder to time.