I plan to travel from

• Frankfurt to IAD in business class purchased using miles/award points.
  
• From IAD to Cleveland, purchased using Cash/Money. This is in first class as no business class seat option available. Either economy or first only.
  

At gate checkin in Frankfurt, can we link the itineraies together, so my bags go all the way to cleveland?

The only way to know for sure is to phone the airline and ask.

A note re "From IAD to Cleveland, purchased using Cash/Money. This is in first class as no business class seat option available. Either economy or first only." That is normal -- on most airlines there is no Domestic 'business class'. It is just Economy and First class. Domestic 'First class' and International Business class while not comparable in service are both at the front of the place. (there are some exceptions on premium long routes like LAX or SFO to New York where there are sometimes three-class set ups)

I mostly fly business transatlantic -- and every domestic connecting leg has been called 'First Class' but was really Business class.

Is this on the same airline?

If on the same airline should be no problem. If on different airlines, need to check with first airline to see if they will do it.

BTW if on the same airline, could you not include the IAD-Cleveland leg on the award redemption?

If your IAD->CLE flight is on United then (almost) certainly yes, Your FRA->IAD nonstop flight is operated by United or Lufthansa, right? They are Star Alliance partners and, more importantly I think, those nonstop flights are all codeshares. If you got the award ticket using United miles then your flkight number is UA 9xx or UA 8xxx, the latter being the ones operated by Lufthansa. I suppose it could be an LH flight number. Or maybe you used Lufthansa miles. Doesn't matter, you can check the bag through to a United flight to CLE.

Call United and see if you can have a CLE flight added to the award itinerary, as Seamus suggested. There might be an economy seat available (actually, before you call you should check online to see if an award seat is available on a CLE flight that connects with your FRA flight).

You can ask now to have the two tickets linked (am not sure if they do this anymore)

" There might be an economy seat available . . . "

I don't think the OP is looking for a Economy ticket. I read it to mean the transatlantic flight is in Business and he tried to book the domestic leg in Business too but it is in First . . . because only First and Economy are offered. Almost certainly, the domestic configuration is two class -- just First Class and Economy (which is essentially Business and Economy)

We can continue to speculate on what OP meant but am not sure it will do any good if the OP never replies to clarify some things.

mjs provided the almost perfect reply to the simple question in the OP. Better would be to ask which airline(s) are involved.

The question was: I have two tickets FRA-IAD and IAD-CLE, can I get my bags checked through from FRA to CLE?

I would bet that the answer is 'yes'. I would also bet that it is because both flights are on United or LH/UA.

"I would bet that the answer is 'yes'."

It totally depends on which airlines are involved. Some absolutely refuse to check bags through on separate tickets even when both tickets are on the same carrier.

>> It totally depends

obviously