Your Blu Ray player will connect to the HDTV via 3 ways. The player most likely included the old standard SD cables ( Yellow, White and Red) to allow you to connect to the any of the TVs standard AV inputs. Some players may have included a HDMI cable that you connect between the player and one of the HDMI inputs on the TV. The only converter box you will need is to tune HD channels that are not broadcast over the air. There is no reason, and in most cases, no way to connect a Blu Ray player through a converter box. they ususlayy send signals to different inputs on a TV.
No you won't need a converter box. This is how it should be setup blu-ray DVD should have a HDMI cable running from blu ray player to TV. You do not need a HDMI cable running from cable box to blu ray player, but if you do want HIGH DEFINITION while watching you're favorite TV show you will need to upgrade your cable box so you can receive the HD channels.
When you do receive your new cable box if you decide to upgrade there will be a port on the box somewhere that looks like a HDMI should go in there you would run the HDMI cable from the new cable box to the TV.
One more thing to be able to get audio from your new blu-ray player you will need to buy a fiber optic cable if your blu-ray supports it. then just connect the fiber optic cable to the back of the blu-ray player to the back of the TV. This is the best way to get your audio. If your blu-ray player doesn't support fiber optic just use any old RCA cables yellow white red connect the yellow to the back of the blu-ray player input and same as red and white.
to hook up the blu ray player setup should go something like this
step 1. plug blu ray player into wall or power surge
step 2. connect the HDMI to the back of the blu ray player
step 3. connect the other end of the HDMI to the back of your high def tv.
step 4 connect rca's or fiber optic
(note: fiber optic is transferred quicker)
step 5. turn on the blu ray player
step 6. your done
http://www.allproducts.com/ee/jh/sc_fiber_optic_patch_cord-l.jpg
^^^This is what a fiber optic cable looks like
To connect the cable box to tv.
step 1. plug in the box
step 2. run the cable from your cable box to the back of your tv.
step 3. if you have a high definition cable box already use another HDMI cable and plug it into the cable box
step 4. plug the other end of the HDMI cable into the back of your tv.
step 5. turn your tv on and cable box
step 6. your done