To rate your play on the round-the-clock game, you could use the MPR stat. This stands for "Marks Per Round" which mean the number of successful hit on a three darts throw. This stat is often used for the "cricket" game. The difference is that in cricket, a double count two marks and a treble, you guess it, three marks. That means that your scores will always be a little lower than on a play of cricket.
All this to say I often play darts on VSPHOENIX machine. These are electronic darts games that are found in bar and arcade rooms. This company also organises tournaments and they have a rating system to class the players. You can find it on this page.If you look at the table on the bottom, you'll see that a "perfect score", which mean an MPR of 3, is a class A15. You can see that this is about in the middle of the table. That mean this is quite a high-score (even more considering you don't count double and treble)!
With your 138 darts to reach the 20, you have an MPR of 0.53 and an N1 ranking. By experience, I would say that this just mean you are a beginner player. To give you a reference, I play one or two times a week since about one year and I normally have a rank around CC5, but I occasionally also get an N1 rank on some game.
If you are really worried about your performance, you could try to measure your variance as pointed in the paper cited by Nicholas V in the comments. To do this, you can go on this page and use the java applet. This will give you a number between 0 and 100. 0 means your reach exactly where you aim and 100 the throws are totally random.
I also recommend you to read _Don't try for triple 20 which is a more casual version of the "A Statistician Plays Darts" paper. You can also read Darts for Geeks: Statistician Cracks the Game’s Secretsthis Wired article which also talks about this project.
As a final word, I would say that the most important in games like darts is not the performance but the fun you have. Try to play with friends which are of the same level as you while you have a drink. Whatever the scores, the fun will be there!